I suspect that the very dispute between collapse and many worlds is an artifact of asking about the behaviour of objective states of the system, and if it is possible to avoid speaking about states, the problem disappears. I want to explain away what MWI proponents want to explain.

Amen to that. Whenever we cease believing we are working with models and doing phenomenology, and start believing we are dealing with reality and doing ontology; at that point we have stopped doing science and entered the realm of metaphysics.

But I thought we already have one [relativistic MWI]. Feynman's sum-over-histories approach.

Could you explain in more detail?

Be forewarned that my physics is at the "QM and QFT for Dummies" level. But I thought that a slogan of "one Everett world = one Feynman diagram" had some validity. At least if you think of really big diagrams. (>5%)