The theory is whatever tells you what you will observe in the real world.
On the case of QM, that's the set of equations, observable operators, and the instructions of hoe to interpret them. None of those predict anything alone.
On the case of QM, that's the set of equations, observable operators, and the instructions of hoe to interpret them. None of those predict anything alone.
Besides, the Schrodinger equation does not predict how the state will evolve after a measurement is done. That's the entire problem.
And that state is a completely made up mathematical artifact.
And yes, the Schrödinger equation predicts exactly how a state evolves, even after a measurement. It does not tell you what the outcome of a measurement is - you give it the outcome, and it gives you the time evolution.
There is a problem, and it lies in what happens during measurement - or what is the exact nature of measurement. But given any state, the SE tells you how it evolves.