> Many worlds is just the Schrödinger equation without the wave function collapse.
It is not, it is the Schrödinger equation plus an extra rule that says that classical observers are only certain kinds of quantum states. There's no explanation in the MWI for why a device never reads out that a particle is 1/sqrt(2) |here> + 1/sqrt(2)|there>, even though that is exactly as probable a state as being |here>. And it turns out that this is exactly equivalent to positing wave function collapse / the Born rule.
And no, decoherence has nothing to do with this. Decoherence explains why there is no observable interference at the classical level between these states, not why you can only observe some states.
This is known as the preferred basis problem, for anyone who wants to read more about it.
Doesn’t the structure of the Hamiltonian (local interactions) set what the preferred basis are?
Well, when we set an experiment up, we can measure in any basis we like, and get the corresponding results as probabilities per the Born rule, so I don't think there are any constraints on this. Plus, local interactions can't explain why the same classical basis arises globally - after all, when we investigate a distant star, it's never detected as being in some other basis then the local area.