Is There a Probability Measure?

I don't think there's anything more to be done in the cause of explaining the mechanism. We did explain the mechanism, self-locating uncertainty when the wave function branches. That's the mechanism. I suspect that your thought that we haven't explained the mechanism is a relic of the thought that if you have probabilities, they have to reduce at some point to literally counting. The thing about many worlds is, let's not over-complicate this. It is crystal 100% superduper clear that if you think that the theory implies any probability rule at all, that probability rule is going to be the born rule.