"The most austere version of QM" is marketing bullshit and not widely accepted (or particularly meaningful). Many worlds does postulate its many worlds rather than predict them as it provides no way their existence can ever be tested for.
Equating Schrodinger's equation with many worlds is both intellectual dishonesty and begging the question. Any interpretation of conventional quantum mechanics involves Schrodinger's equation. Penrose's ideas (referenced by Sean Carroll in quotes others have posted on this thread) involve violations of Schrodinger's equation because they are actual new theories that differ from standard quantum mechanics. In other words, Penrose is doing actual physics rather than blowing smoke up people's arses. This fact doesn't convey some falsifiability on many-worlds, which is just one of several interpretations of QM that all make the same predictions.
Many Worlds just postulates that if you put a cat in a superposition of two quantum states (alive + dead), there is no sensation associated with that superposition. The cat is in a superposition of having the sensation of being alive, and having the sensation of being dead, but there is no observable quality of "being in a superposition of alive and dead". Both of the superimposed pure states of the cat feel decidedly one way or the other.
Hence, when a human is in a superposition, we would not know it. Both of our superimposed states are experiencing the feeling of being in a pure state of looking at an instrument measuring a photon with polarization ↑, or looking at an instrument measuring a photon with polarization ↓. Being in an (↑ + ↓) superposition doesn't feel like looking at an instrument display and seeing a blurry reading. We feel like we saw the instrument reading ↑, and we (as an element of the quantum system) cannot interact with the branch of the wavefunction that we are superimposed with in which we saw the instrument reading ↓.
This isn't a separate postulate, it's just straightforward quantum theory applied to quantum systems large enough to have opinions about their own state.
Your conclusion does not follow from your premise. MWI does postulate different stuff from Copenhagen, but the existence of many worlds is not one of them. Just because I postulate Peano arithmetic doesn’t mean I also postulate 2+2=4. The latter is but a consequence of the former.
My understanding of MWI, is that it just "believes" the equations. Specifically, if the equations say the complex amplitude is not null, then this stuff is real. And that’s about it. It’s not our fault the equations describe non-null amplitudes, that if real are many worlds.
This interpretation business could apply to macro-scale physics as well: when you send a probe so far out there that it crosses the limit of the observable universe, does it cease to exist? What if there’s people inside, will they die? One would be hard pressed to argue from their discontinued existence: why would we disbelieve the equations of relativity, that work so well for everything we can observe?
Conversely, what makes you think that the equations that work so well on everything we can observe, would somehow stop working (or stop applying) on the stuff we cannot observe? I know you didn’t explicitly disbelieve the equations, but since we can’t at the same time believe in their general applicability and disbelieve MWI, it’s hard to interpret your refusal to acknowledge MIW as the most probably hypothesis as anything but scepticism about the applicability of those equations.
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To throw you a bone I personally do have a reason to be wary: physics is currently inconsistent, and when we get a credible theory of everything, it may very well use different equations that do not describe many worlds, make long range probes vanish, or both.
Of course they cease to exist as do the people ... just like sailing ships and sailors did when they travelled past the visible horizon.
This isn't the conundrum you're looking for.
“Worlds” is inaccurate since there is still only one world (the ordinary, uncontroversial, tensor product Hilbert space of textbook qm). It the stuff in the world that enters into superposition, among which stuff are the atoms making up “observers”.
Similarly “completion” would be a better word than “interpretation”.
However, the absurdity of this consequence should mot lead us to doubt the existence of motion, but quite the opposite, it hints to us that the either the postulate is false, or otherwise we are missing some essential knowledge that allows to determine the correct consequences of that postulate.
Also, MWI doesn't assume infinite worlds, it predicts them. It's a consequence, not a postulate.