There are two issues here: many worlds, and the alleged desirability or necessity of abandoning continuity of physical existence as a criterion of identity, whether physical or personal.

Regarding many worlds, I will put it this way. There are several specific proposals out there claiming to derive the Born probabilities. Pick one, and I will tell you what's wrong with it. Without the probabilities, you are simply saying "all worlds exist, this is one of them, details to come".

Regarding "continuity of substance" versus "identity of substance"... If I was seriously going to maintain the view I suggested - that encapsulated local entanglements permit a notion of persistence in time - then I would try to reconceptualize the physics so that identity of substance applied. What was formerly described as three entangled particles, I would want to describe as one thing with a big and evolving state.