The idea of the trope is that at some point, Earth is doomed, humans flee into the stars, knowledge of Earth is lost in the process, eventually becoming legend or myth, either due to loss of information, revisionism due to state propaganda, or even something gone horribly wrong in the early days.
So, the question is: in reality, is this trope a viable scenario (even if temporary) or is it unrealistic from the start?
I think it is very plausible, even if Earth still exists and is still inhabited, at least if the "no faster than light travel/communication" still stands.
The closest known terrestrial exoplanet is Proxima Centauri b, which is 4.22 light years away. This means, every communication has a round trip time of about 8.5 years! Now imagine, that Proxima Centauri b is not inhabitable and maybe the next few planets aren't either, so maybe our colonists settle on one if the TRAPPIST planets. They are about 40 light years away. So every communication takes a whole lifetime until an answer arrives. Then there is the question, if it is even possible to communicate over such long distances at all.
So, no matter which extrasolar planet will be the home of our settlers, communication and transportation between Earth and our settlers' new home will be extremely difficult if at all possible.
Now, I guess, it will not take long until the regular population will not know more than myths about Earth anymore. Most people are extremely ignorant towards places they have never been to. Just compare it with the average knowledge about any far-away place. For example, a lot of Europeans don't even know that there is an African Union, that, similar to the European Union tries to unify the African countries into a larger construct. Likewise many Americans don't know that Europe consists of many countries. And that's just superficial detail, nothing more.
Same goes for times past. So many people think, the dark ages lasted until the 1800s or believe that people in the dark ages thought the Earth was flat.
For historians to loose that kind of information without a major event (e.g. data loss) it will probably take quite long, if it would happen at all. But in most stories involving the myth of "Earth that was" the protagonists are not historians but some random people. So it is very plausible, that they would not know anything more about "Earth that was" than regular people now know about the life under the rule of the Babenbergers (they ruled some areas in Central Europe where I am from from 976 until 1246, and I have no clue as to how life was like for a regular person at that time).
One other point that might be also relevant is what kind of information the historians took with them from Earth. Depending on where you are from, history is taught with a very different focus. If you live in the USA, you learn a lot about the American civil war. If you live in Europe, the American civil war is hardly more than a footnote, while the French revolution is really important. If you live in Japan you will hear a lot about the Sengoku period. And in none of these countries you will learn a lot about the history of Zimbabwe. Especially if these settlers hailed from a totalitarian or very patriotic country their knowledge about "Earth that was" might be very distorted and incomplete.
Plausible scenario:
... After a millennia of traveling by the single colony ship to escape Earths doom a catastrophic error occurred in the ships computer system causing the digital loss of star charts and Earth's known location.
Kind of hard to figure out where Earth is if you don't remember your flight path and accrued solar data.
I feel like you have to start with a discussion of the exodus and when you start to think about it on the surface seems like a crazy, horribly unrealistic option.
But what other options are there? The death of the species seems like the only one to me.
With that in mind we can make it plausible. With extinction the alternative is there any end we wouldn't go to so avoid it?
Some points:
Losing track of earth in such a scenario isn't that difficult. Particularly if the technology available is just barely sufficient for the task...which seems like the situation we humans would put ourselves in...
The time it takes to travel to distant stars (in a physically consistent with reality way) means anything can happen with information. We convince ourselves of things that aren't true on a nearly daily basis due to feelings...losing the reality of Earth when you can't even see it doesn't seem all that hard.
All in all I would say it is a completely viable scenario, and frankly the most likely, maybe not today but certainly in the future. From a story telling perspective I would say it obviously works.
EVE Online's prologue uses this trope, and has some clever ways to make it a bit more plausible. It's all laid out in their old intro video. Here's the summary...
Humanity has "outgrown Earth" and go out in a "desperate quest to colonize other worlds". Humanity builds FTL stargates to jump between worlds. Eventually even the stargates could not take them further (this could mean humanity covers the whole Milky Way galaxy). With nowhere to expand to, humanity begins fighting itself. Your basic "Earth Is Doomed" story, but it's the whole galaxy.
Then Earth discovers the "EVE Gate", a wormhole to another part of the universe called "New Eden". It's so far away they're not sure where it is in relation to Earth. Millions of colonists rush through and eventually settle thousands of new worlds.
Then one day, without warning, the gate collapses leaving thousands of colonies suddenly cut off from Earth and their supply line to Earth.
Lacking the ability to build, maintain, and repair space ships and jump gates without support from the Milky Way, each colony lost contact with the others for thousands of years. Most colonies are not yet self-sustaining nor are their planets even fully terraformed. Cut off from each other, these colonies die. The surviving isolated and incomplete colonies descend into anarchy and barbarism. This begins "a dark age that erases civilization as it is known to be, as the accumulated knowledge of millenia slowly eroded".
Finally two planets independently reinvent space and FTL travel, and humanity begins building stellar empires again. But the dark age has left "the memories of our past transformed into legend and myth".
EVE Online solves the problems with the "Earth That Was" in a few clever ways.
First, New Eden is so far away from Earth that even "the ancients" didn't know where it was in relation to Earth. One explanation is it is within the same Universe, but outside the universe observable from Earth. This neatly solves the problem of how they lost knowledge of where Earth is: they didn't know it in the first place.
Second, the only way to get to New Eden is via the EVE Gate wormhole. This leaves no breadcrumbs to follow back to Earth. There's no metaphorical "road" nor string of colonies nor evidence of space campsites to follow back to Earth as we can with, for example, Polynesian expansion. Even if they wanted to, the residents of New Eden can't trace the steps of the ancients. The ancients can't do it either.
Third, FTL in New Eden is limited. They model distances in game between planets and star systems accurately. The FTL drive in ships is sufficient to quickly move from planet to planet within a star system, they are far, far too slow to cover interstellar distances. For example, the distance from Earth to Saturn is about 1 billion km, give or take. The distance to Alpha Centauri, the nearest star system, is 25,000 billon km; 20,000 times further.
Most ships need established jump gates to travel between star systems. These represent an enormous investment in time and resources to establish and maintain, something struggling colonies could not afford. While there are ships with their own interstellar jump drives, they are enormous, expensive, fuel hungry, and limited in range. This makes casual interplanetary exploration difficult and expensive. Getting to the neighboring star system requires either an established jump gate, or an enormous investment in a jump capable capital ship, or a lot of patience. This reduces the ability for humanity to explore making it more plausible they can't find Earth again. Firefly does something similar with "The Verse" being just one very large and heavily modified star system.
And just to cover any holes, the "dark age" of New Eden is so much worse and complete and sudden than anything humanity has ever seen. In examples from Earth history, there's always some way communications or trade can happen, even if it's across a perilous ocean or desert. In New Eden, the loss of space flight leaves all the colonies cut off from each other by the cold vacuum of space. On top of that, most colonies die off, only a handful survive. Those survivors cannot communicate with other survivors for millennia.
This hard information and trade gap between colonies, plus the swiftness and suddenness of the EVE Gate's collapse, makes the New Eden dark age lengthy and complete. It is far more plausible that all history and technology would be wiped out with a millennia of isolation in marginal environments.
This long, complete dark age also allows EVE Online to reset the timeline of humanity. EVE Online is set millennia in the future. On this scale, humanity and its technology would be incomprehensible to a 21st century observer. It would be difficult to explain why we have the Space Spanish Inquisition fighting the Space French.
Instead, the New Eden dark age allows them to be divergent cultures emerging from their own isolated post-apocalyptic planets. Their sometimes primitive and regressive cultures, for example there is slavery in New Eden, can be explained because each one is the result of humanity being knocked back to basic survival in isolation. Each in their own Darwinian struggle on their own little Galápagos.
The closest analogy to New Eden from Earth's history might be Easter Island. Settled around 700 AD, geographically distant from any other inhabited island, it became completely isolated around 1500 AD. By the time Europeans arrived in 1722 it had consumed most of its resources and lost 80% of its population.
The question makes more sense if we think of it in mytho-poetic terms than hard loss of navigation data.
Where was that place again....
Consider your own memories. Unless something truly traumatic happened in your school or childhood, you probably have a fairly idealic recall of that period of your life. Now go even farther back, to your family history. The stories passed down the generations have been selectively edited. Great grandfather is the hero of the story, or the "flood of '21" takes on mythological proportions. Careful research into historical records might not even reveal your Great Grandfather's name, and the rainfall in '21 might have been somewhat higher than average....
Going even farther, look at national mythology. America is "The Shining City on the Hill", Britannia "Rules the Waves", Moscow is"The Third Rome", China is "The Middle Kingdom" between Heaven and Earth and so on. As a personal recollection, I was serving in Bosnia and being told the story of the "Battle of Kosovo Polje" by local Serbs in such vivid detail that you'd imagine it was fought in the 1990's during the civil wars. It took place in 1389
I thought this took place just a few years ago....
So while everyone may actually be able to point to Sol from their position in space, referenced between thirteen different millisecond pulsars, the actual remembrance of events and places on Earth will be so coloured and distorted that if you or I were to step out of a wormhole and have the tales of Earth recounted to us, we would come away convinced that these people came from an alien planet.
Directions to Earth
So the "Earth That Was" may be a clear memory and starting point of the colony's history, but may have little to do with the actual Earth itself.
The timeline looks something like this:
Now let's look at how that might happen.
Loss of contact
This is far and away the easiest one. Some catacylsm happens on Earth that wipes out most/all human life and/or all advanced technology that might be used to talk to our colony.
Assuming that FTL communication is possible (subspace, etc) then attempts would still be made by the colony, but with no response eventually the program could be shut down as resources dwindled. Assuming no FTL communication, then at some point all radio chatter from Earth would stop. Our constant stream of broadcasts would be cut off. In that scenario, the colony can pretty readily guess that Earth isn't around anymore.
Loss of knowledge
The colony needs to be in a very specific state of development for this scenario to be truly feasible: it needs to be independent enough that survival without a supply line from Earth is possible, but dependent enough that most critical systems are irreplaceable.
Similarly, having a large, diverse, and specialized population of civilians would be helpful in this case. One unified leader, whether scientific, religious, or military, could command enough authority to maintain order. Whereas many different groups with no clear central authority could easily descend into chaos, resulting in the loss of computer systems and computer data. As these systems fail without replacements, the people who know how to maintain and fix them become overwhelmed. Without years of schooling to teach these highly specialized skills, because everyone needs to help out on the farm, each subsequent trainee knows less and less about these advanced systems. And as the specialized knowledge dies out, the systems fail completely.
Perpetuate this state for at least one and a half generations and all you've got to go on is your aging great-grandfather's half-remembered paper sketch of a star chart.
Passing into legend
Humans have been pretty obsessed with their origin since we started thinking about things. We now know of course that biologically we evolved in North Western Africa and the earliest signs of non-nomadic human civilizations were in the Fertile Crescent of the Middle East. There is no reason to believe that a technologically regressed interstellar colony would be any less obsessed with passing on stories of Earth.
As the generations press on, these stories go through a cosmic game of Telephone until suddenly airplanes are flying cities, urban metropolises are cities with towers that pierced the sky, and we could fly through the air just as well as we could fly through the water.
Denial of Earth's existence
This one is a bit tricky. The best way to accomplish this would be to have some religious or social movement take primary control of the colony that denies the existence of Earth as a matter of principle/philosophy/faith. Any scientist or historian will very easily be able to prove that humanity didn't come from Here, so maybe we did come from the mythical Earth.
The best bet of having a society deny it entirely is to make the denial based on some unprovable element, something that can't be proven or disproven via scientific analysis.
Depending on what assumptions you make about the progress of technology, it may not only be plausible but relatively likely. Its easiest to look at the steps in order.
1. Earth is doomed
Over a long enough timespan, this may be almost certain. There is a better chance than most people want to think about that we will make this world uninhabitable or at least undesirable in the relatively near future through pollution, climate change, or nuclear war. There are a number of extinction-level natural events which are possible. Of course, sufficient technology with sufficient forewarning should allow us to prevent or mitigate most of those, but it is plausible that one will occur that cannot be prevented. And of course, over a long enough time span the sun will eventually change to make the solar system uninhabitable.
2. humans flee into the stars,
How likely this is in reality is hard to say because it depends on whether it is even possible to develop the technology to travel faster than light or create something akin to suspended animation or create a generation ship capable of lasting for the entire trip to another habitable planet. But while it is hard to say how likely those things are, they are certainly plausible and absolutely ubiquitous in science fiction.
If we develop any of those technologies, then it is almost certain we will attempt to establish colonies on other planets long before earth becomes doomed, especially if we see that doom coming.
3. knowledge of Earth is lost in the process, eventually becoming legend or myth
If we accept parts 1 & 2, then over a long enough time span this is all but certain. If you look back at ancient or even medieval history much of what we think we know is dubious or disputed and almost certainly exaggerated.
You might think that our hypothetical settlers that abandoned earth will avoid having that issue with earth because we have better record keeping now than they did in the ancient or medieval worlds, but time has a tendency of twisting such things. Records will be lost. Storage media will be corrupted. Other storage media may be fine but people may forget how to read it.
Worse, people will deliberately destroy or alter records for all kinds of reasons. People will outright twist the truth and lie about what is going on when they make the records. Even if accurate records are mixed in with the lies, it can be hard to know which ones to trust and which not to. Superior record keeping may mean that it will take longer for the truth to be reduced to myth and legend, but it will almost certainly happen.
And those are just the simple historical processes that occur with the passage of time. In our hypothetical, our settlers abandoned a dying planet. They may be doing so with limits to what can be carried and without putting a high priority on bringing the historical records with them.
And all of that is without positing something plausible but less likely such as the rise of a fanatical religion that wants to take coordinated action to discredit knowledge of earth.
In short, the entire sequence if definitely plausible so long as we take the assumption that space colonization is plausible at all.
I think it's extremely plausible because of how difficult space travel is. (And probably always will be). And of course, how hard it will be to have a conversation with 'home'.
A colony ship will take multiple years, potentially multiple generations to reach it's destination. Over that time, even when staffed with the very best scientists, you'll have an inevitable reduction in knowledge/education level. Things will break aboard your colony ship, and they simply won't be possible to repair - even with excellent electronics skills and a good soldering iron, manufacturing replacement microprocessors will be extremely hard.
And once you reach the colony, then ... maintaining the kind of advanced technology that we're all used to is going to be impossible initially. With 'just' a few thousand colonists on a brand new settlement, I think a reversion to a basic agricultural lifestyle would be largely inevitable.
Once that happens, it only takes one accident over the course of multiple decades to wipe out the records. Say, the 'book store' gets flooded, or there's a solar storm and the backups get wiped, or ... well, all sorts of things.
It's really not that unusual when doing backups to find a proportion of your restores just fail if you've not been checking them regularly, and as the specialist knowledge faded over generations then that problem would get worse.
