[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.