You're right. It's a little off, because your example earlier shows that subalgebras in the algebraic formulation do not in general correspond to subsystems in the Hilbert space formulation. I don't know what I should be asking instead, so I'll just say that nothing that I know of gives me any reason to think that the problems I've been talking about can be avoided by using the algebraic approach to QM. We're going to need the Born rule or something equivalent to it, no matter what approach we're using, and its main purpose (in the MWI) is to define ways to decompose the "omnium" into subsystems.Your question seems a little off
Each decomposition defines a specific way to describe the unitary time evolution of the state vector of the universe in terms of correlations between subsystems. (This is a lot like how each inertial frame in SR gives us a specific way to describe what's going on in the universe).