The wave equation of QM is just a differential equation too. The interpretation of what its values actually are is quite contentious AFAIU, and what the Born rule used to derive the probabilities "means" depends on the interpretation.

But yes, if you have "collapsed the wave function", you don't have the information to "backtrack" anymore. But I don't think the information paradox is about this (although unsure why). AFAIU one problem is that black holes throw out even the statistical information available in the wavefunction.