This is a creative idea, and I would love to see how it turns out. Let's play with it until it works.
Scientific problems and solutions to them
under the ice of Europa
I would caution you with this one. Europa is incredibly cold, so any organisms developing will want to make use of thermal energy below the surface - such as geysers. Consider less of an "all encompassing" creature and more of a web across all geologically active faults and spanning all geysers. Somewhat hard to do, but possible.
However, unlike on Earth, this single progenitor organism never died out or was outcompeted by its offspring
Not necessary to provide the circumstances you desire, and it adds a layer of unlikeliness. Consider killing it anyways, I'm sure its offspring will suffice.
eventually it even began to group together with other cells of its kind to form simple logic gates (kind of like what we've been doing in the lab). One thing led to another and this ancestral species eventually developed into a bacterial supercomputer, from which emerged crude sentience.
This isn't impossible, but as other answers have pointed out, it's also not very likely. Consider something like that described in the second best answer on the recent question you asked, which involves "smart" colonies of algae. While I'm not sure exactly how this evolves, Earth examples prove it's completely possible.
As other answers noted, crude sentience must exist for a purpose - if this organism survives by reaching heat and processing minerals, which can be done through plantlike reflexes, there is no need to evolve higher thinking. Consider evolving a couple predators to refine those reflexes over time.
As the first and as far as it knew only intelligence underneath the icy surface of Europa
See the above - there must be other intelligences to co-evolve if you want a crude sense of consciousness to develop.
Also note that reproduction happens - while this may be one big clump of "algae", with individual cells that reproduce asexually, some are bound to split off and attempt to colonize. You must explain why there are not more of this organism.
You could solve this problem by making your colony analogous to Pando, the largest single tree on Earth. It's beneficial for cells to remain with the main "mind" because it provides nourishment; if they split off, they will be devoured by the newly evolving plankton and other microorganisms.
it inevitably came to the conclusion that all these other more complex organisms that it birthed so long ago lacked its collective hive intelligence, or indeed any intelligence, and were there merely to serve as extensions of itself
I disagree with "inevitable" and I have discussed why there must be other intelligent (though not necessarily self-aware) organisms above. However, the "extensions of itself" idea is somewhat like the original "animals are here to serve humans" as opposed to "they evolved too" mindset, so it is credible.
Relying on their distant evolutionary connection, it began to infiltrate other organisms in its ecosystem and turn them into biological fingerpuppets, as they were to it little more than large colonies of cells marginally differentiated from its own that existed only to serve a specific purpose, like a new limb.
The evolutionary connection part isn't credible - we are connected to primates, but does that mean we can control their minds with ours? I suggest leaving that out.
You could, however, combine some qualities of angler fish - in which the males fuse with the females to reproduce - and of a certain nasty fungus Ophiocordyceps unilateralis - which has evolved to control the brains of ants at a fundamental level. Perhaps the motivation for this is that new bodies provide nutrients and processing power - but "mind pry" thing will take time to evolve, something which an asexual colony will not be able to do for a very, very long time.
The marginal differentiation part may be important, but would involve creating more mouths to feed with little productivity; dead weight will become a problem. I suggest absorbing much different intelligent species with brains that are of some use - or just absorbing to digest - rather than to add mass.
At the end of the process, what we have is a single massive colony of this primordial microorganism that has gained intelligence and has differentiated its cells into many different types which each form their own colonies (read: organisms/animals)
I've addressed most of what can allow this to happen by now - the "different types" would come from capturing different organisms to use, etc. Sounds good.
central hive intelligence
As TessellatingHeckler pointed out, brains are not fast when considering the size of this thing - so a creature spanning the planet will have some delays. Combining slow metabolism (most of Europa is cold) and long latency issues, expect a fairly slow mind.
Regional intelligences with some overlap might work better - they can all do the same things, but memories may not be consistent, and there will be a delay in spreading any new.
As a character
If the mind is a character instead of an obstacle, it will be extremely complex, and you could create an infinite number of plots around it. The ever-present dementia, confusion, and isolation may be aspects of its personality, a cause for potential suffering, and a reason to sympathize/empathize with it.
As a monster
On the flip side, if it's "evil" or "heartless" and functions just for its own survival, your characters may take advantage of latency and confusion to escape it or destroy it.
As a philosophical discussion driver
I urge you to take advantage of the potential discussions this creature offers you. To list a few:
- Who defines good and evil?
- Perhaps this hive mind, growing up alone, isolated, and hungry, will have different morals. Who are we to decide that it's morals are "wrong"?
- What are the differences between living and being alive?
- Similar to the above. Sure, its "living" in the biological sense, but can this creature really feel "alive", in constant solitude, cold, and hunger?
- What happens when you die?
- A scared, isolated mind may witness the death around it and come up with a religious viewpoint.
- If it has consistent views with ours, perhaps it proves there is a creator.
- If it has completely different (but not disprovable) ideas, are they any less valid than ours?
- Do we have a soul; are we special?
- Is there a fundamental difference between human consciousness and others out there?
- Is "soul" just an idea, or do we believe after meeting this that we're special?
Other notes and discussion
Hopefully I've brought you close to hard sci-fi, but I can't guarantee anything. Speculative evolution is difficult to call "hard" but I think you're close(ish) now
is it complete garbage psuedoscience
Even if this ends up being pseudoscientific, there's no reason to stick to hardness! The facts and figures explaining a creature can be boring - imagine if Godzilla explained how the creature dealt with crushing gravity! Furthermore, I'm sure your story will be great regardless of how realistic it is.
This is an awesome idea, can be justified with a few tweaks, and would make an excellent, complex story that reaches deeper levels of thought. Go for it, dude.