The Universe is already the sum total of all the time travels to ever happen.
This is an alternate, more intuitive way of stating that the probability of changing the past is zero. It is zero because the past has already been changed. Let's see how this plays out with a few apparent paradoxes. Of course, a chronophysicist will tell you that this is a gross oversimplification, the way the Bohr model is for an atomic physicist.
- The inventor of the time machine goes back in time to give their past self the plans
At some point, in a hypothetical not-yet-time-travelled universe, someone invents time travel. The decide that spending decades to do so wasn't worth it, so they travel in the past to give the plans to their young past self who just started working on the problem, creating a hypothetical time-travelled-once universe. The new inventor builds a machine in a few months and decides to go back in time to give the plans to their much younger teenage self, to skip some more years, etc... Until at some point, they went back in time and gave the plans back at the exact time where they received it.
All those previous hypothetical universes don't really exist, only the "final" one does. They are theoretical constructs that help calculating the actual result, not unlike virtual particles in quantum mechanics.
And voilà, you end up with what appears to be a causal loop at first glance, but actually isn't.
- Grandfather paradox redux: after receiving the plans in their youth, the inventor decides to not go back in time and give their past self the plans
This is impossible in the real universe, because it already happened. The inventor, or some very convincing fake, already went back in time to give the plans. So if the inventor didn't, then something else did, disguising itself as the inventor. How improbable this is depends on what is going on in the entire timeline from this point - as a rule of thumb, it's not going to happen without a good reason.
So in most cases, it won't be a "final" state. Then, how does the next hypothetical timeline iteration happens? The trick is, the new looping event doesn't have to be the same. Maybe someone else uses the time machine at some point to go back further in time, causing a new event cascade that ends up causing the inventor to not bother with time travel in the first place. Or someone made a genesis bomb, which went back to Big Bang times and scrambled what would become this part of the Universe, causing completely different galaxies to be born.
Side note: Time travel and the Fermi paradox: if the Universe allows technical civilisations to build time machines that don't require a receptor (think H.G. Wells' Time Machine, instead of wormholes), then it allows for said technical civilisations to reach back to deep time, potentially the beginnings of the Universe, and scramble the constituent parts of their future star systems or galaxies to non-existence - whether by accident or self-destructive action is unimportant. This may considerably limit the probability of existence of such civilisation. The good news is, if the NSFP applies, you already live in a universe where your civilisation don't retroactively wipe itself out of existence! Lucky you. Though it may mean your civilisation is probably not going to discover those time machines at all.
So hypothetical timelines will keep modifying until one is reached where its own time travels cause its own events - and this is what actually happens. What looks like a strap-on causal loop is actually the first consistent timeline that could have existed in the first place.
- Causal oscillations: timeline A causes timeline B, timeline B causes timeline A
Alice and Bob play Heads or Tails. Alice says Heads, and Bob reveals that it was Tails. But Alice is cheating: she uses a time machine to tell her past self the result. So what actually happens is that she says Tails. Bub Bob is cheating: he changes the result depending on what Alice says. So what actually happens is that he reveals Heads. But then, it means Alice actually told her past self that it was Heads...
In reality, the oscillation are never perfectly stable. Hypothetical timeline 1 is not quite the same than hypothetical timeline 3, even if in both cases, Alice and Bob choose the same faces in both. Those variations will at some point break the loop and end up with a stable timeline.
What it means for time travellers:
It is not that time travellers cannot change the past, so much that they did already.
While there are what looks like strap-on causal loops, they are actually the natural result of time travels to happen during the existence of the Universe.
And while there may very occasionally be events that look causally inconsistent (that is, caused by the arrival of a time travel that doesn't seem to happen at its supposed departure spacetime), this is caused by a mistake on what is actually the cause of the event (the actual departure spacetime). Note that this can be weaponised: you can change the past (or future) as long as you are not certain what actually happened. This may imply deception of your past self, so an event you think actually happened (preventing you from changing it) did not in fact happen (so you can change what did happen).
Harder to identify but present nonetheless are events that should be more common than they are, like time travel WMD. Those are events that could later cause themselves to not happen, which by nature will proportionally decrease their actual probability.