We understand why the evolution of classical systems is governed by classical probabilities---just churn through the quantum mechanics. Decoherence is very simple...

Decoherence isn't actually enough to show why quantum amplitudes show up as classical probabilities- if it were the Born-amplitude problem in many worlds would be solved. You need assumptions to turn "the wavefunction looks like this" into "the wavefunction looks like this so we expect to see result A with probability whatever." Decoherence tells us off-diagonal elements in the density matrix aren't likely to survive interaction with a larger system- thats not enough to connect to experimental values.

You jump straight from decoherence to "experiencing branches" without defining what you mean by "branch. "