Yes, the MWI is falsifiable. It asserts that objective collapse does not occur, therefore any observation of objective collapse (such as predicted by GRW or Penrose-Diosi) would falsify it.
That's not true falsifiability; its asserting a negative.
I think people resort to MWI because they think it explains everything neatly; it does not!
For example, from my perspective, it does not explain what world I end up in, and if you are saying it's random, you need to come with a fundamental theory of randomness, unless the response is: it just exists, deal with it.
A negative, that can be falsified by a single black swan.
You are mixing up another kind of argument. Claiming there are elephants in the room we cannot see or touch is an example of an unfalsifiable claim.
In fact, claiming there is a quantum collapse, which always looks just like the field equations without a quantum collapse, would make collapse an unfalsifiable theory.
(I don’t believe most proponents of collapse are making that claim.)
