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Well, under Many Worlds, wave function collapse isn't a real "thing"; it's just an illusion caused by the observer becoming entangled with the wave function. Objective Collapse theories, however, propose a real physical mechanism of wave function collapse. If that's true, and there was found to be a real mechanism of collapse, then MW would be impossible, because the wave function would collapse before any "branching" could happen.
And what is there to stop the collapse from being the branch point? In one world, it collapses one way; in another, another. There doesn't seem to be any inconsistency there.
Well, because under Many Worlds, the wave-function not collapsing is the reason there are multiple branches; the wave function is the multiverse. So if the wave function has collapsed into a single, definitive state, then there is only a single, definitive universe.
Sorry, that doesn't prove that there's not actually Many Worlds out there. The whole point is that there would be a single, definitive universe state for every possible valid configuration after wave-function collapse. The reason it's unfalsifiable is that it cannot be proven currently whether or not it's a literal plurality of alternate worlds. I would also argue that if there's but one "definitive universe" state then it's not really a Many Worlds theory at all, but just a different theory of the Universe.
I'm not saying you're wrong, or that this interpretation of Many Worlds is wrong - I'm just saying we've not yet developed a way to prove it one way or another. And if we did develop that technology to prove it one way or another, that would in itself unlock a whole new world of questions to answer. Thinking about what those questions might be is worthwhile science, in my view.