[eigenket]

I am not aware of any interpretation that has the Born rule without assuming something equivalent to it. For example de Broglie–Bohm theory requires you to assume the original distribution of the particles follows the Born rule. QBists just postulate it, consistent histories just postulates it etc.

I am not particularly a many-world proponent, but I do not think it is fair to level this accusation as an issue for many worlds without bringing up that every other interpretation has the same "flaw".

[Jweb_Guru]

AFAIK it can in fact be derived through Gleason's Theorem under the assumption of noncontextuality, so I don't think it's fair to say that nobody can derive the Born rule without assuming it (many people have issues with noncontextuality but that's very much a philosophical thing). The thing you have to demonstrate is that a probability measure actually connects to physical observables in some way, and this is the part that is difficult (and as far as I can tell MWI does nothing to resolve this conundrum).

[eigenket]

Gleason's theorem also makes a big assumption when you require that the measurement outcomes are associated with POVM elements (or projection operators if you don't like POVMs). I lumped that in with "assuming something equivalent to it" since Gleason's theorem (at least by my understanding) is exactly the statement that assuming non-contextuality+POVMs/POMs is equivalent to assuming Born's rule.

Although its really cool I don't think Gleason helps you tie any particular interpretation to the Born rule, since you still have to make a jump to tie your measurement outcome to a POM/POVM element.

As far as your last sentence goes, this is sort of what I was trying to argue in my comment above. The "part that is difficult" that you identify as being unresolved by MWI is also completely unresolved by pilot wave theory, or qbism or consistent histories or any other interpretation (as far as I am aware).

[Jweb_Guru]

I think we're in agreement there (though IMO it's highly nonobvious that noncontextuality+POVMs automatically get you the Born rule, so I don't think it's "cheating" to assume that--obviously any set of axioms that let you derive Born will have such a property!). I was mostly saying, I don't think MWI helps us understand where the probabilities come from any more than any other interpretation--you need something more. And if you can't identify where the probabilities come from, then saying your theory is "deterministic" rings fairly hollow to me.