Bell's Theorem states that there exists no local hidden variable theory for QM. AFAIK that implies that QM must be non-local because the other possibilities of a local theory can be reduced to a hidden variable theory or have no evidence for them.
Not true. Bell's result precludes local realism. So physics is either not local or lacks realism (counterfactual definiteness). The Many Worlds Interpretation keeps the locality but lacks counterfactual definiteness. The fact that MWI is the only interpretation that manages to hold onto both determinism and locality is what makes it so aesthetically pleasing.
Many-worlds still has nonlocal entanglement, IIRC, and all the weird phenomena therein.
It does not require FTL signalling for the entanglement to work.
Care to make rigorous how other interpretations require FTL signaling? For that matter, can you actually rigorously explain what it means for worlds to "branch", or why Born probabilities should be interpreted as though they are probabilities in MWI? Saying "all the possibilities actually happen!" doesn't really explain the correlations we actually observe in any interesting way.
You make some good points. Regarding the Born probabilities, the extrapolation of Schroedinger equation to the whole universe seems to make the concept of probability superfluous and the Born rules lose their sense.