You're correct it is not just a MW problem - making the Born rule an assumption, as in Copenhagen, hardly seems to solve the issue.I don't want to get too off-topic, but just to respond to this: I think Vaidman's view is sufficient. You assume that Hilbert space represents a density of worlds and the only sensible (non-contextual) choice for a measure on Hilbert space is the amplitude squared . Then with some reasonable model of an observer, it's possible to show the observer will experience outcomes according to the Born rule.
Maybe you call that circular because it requires some assumption, but I don't see why that's a MW issue; it seems like most derivations of the Born rule apply to most interpretations. For example, The measurement postulates of quantum mechanics are operationally redundant (2019).
Vaidman's solution seems very similar, or even identical, to Everett's?