One outcome being more probable than another in MW can be interpreted as that outcome being more likely to have occurred in your past. That is also what the frequentist example is measuring, not which outcome is more likely but which outcome is more likely to be in your history.

There is also Quantum Bayseianism in which an outcome being more likely is due to that state being better at copying itself to neighboring states.

In all interpretations (including Copenhagen) the mapping from the unobservable wavefunction is taken as axiom (well, one could argue that Relational QM/Transactional interpretation avoids this, but just adds another axiom).