> 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.