> I see no implications of collapse with this -- just probabilities of what values you will obtain in experiment.

Not exactly, your statement is correct for single states, however it breaks down for entangled states. When one wavefunction describes two or more particle a measurement on one particle can be thought of collapsing the two particle wavefunction, which then forces the other particle into a well defined state.

Edit, I am not arguing in favour many worlds theory, only that the 'collapse' of the wavefunction is a well developed and useful idea in quantum physics.