To my mind, the micro assumptions don't change the macro conclusions, they only change the way we talk about and justify them.

I agree with you that one should reach most if not all of the same conclusions from a strict Newtonian perspective (or from a Copenhagenite perspective, and so on). But the way it's talked about does scare me, because it's difficult for me to tell why they believe the things they believe, and opaque reasoning rings several warning bells.

That is, to answer your original question- "Is this really the case?"- it certainly is the case that it appears that EY's philosophy builds directly on his interpretation of QM. When judging by appearances, we have to take the language into account, and to go deeper requires that you go down the rabbit hole to tell whether or not EY's philosophy actually requires those things- and that rabbit hole is one that is forbidding for non-mathematicians and oddly disquieting for physicists (at least, that's my impression as a physicist). QM is an inferential distance minefield.

It seems to me that MWI is just a convenient visualization trick, and thus there is equivalence, but I don't feel I understand EY's philosophy and its development well enough to argue for that interpretation.