The appearance of the Born Rule is explained by decoherence + bohmian mechanics or decoherence + many worlds. (At least my interpretation of many worlds, which is an ensemble of Bohmian trajectories. There are other interpretations of many worlds which I don't understand well.)

The actual Born rule is approximately explained by decoherence + GRW.

So decoherence + simple ontology is a great way to avoid the vagueness that Copenhagen gives you.