[Something else about inteepretarions] We define the wavefunction as something that contains all the information we can learn about the quantum state. After reading the PSE answers about probability in quantum
mechanics, I am tempted to think that a hermitian operator $\hat{M }$ kind of filter out the probability amplitude that corresponds to the information of the observable $m$ only, which then, when the overlap of that with the wavefunction is computed via the inner product $\langle \psi \lvert \hat{M} \rvert \psi \rangle$
with give the sum of all values of $m$ weighted by the amplitude squared (i.e probability), thus giving the expectation value as expected. However there is an obvious question to ask. We knew that
while there are a lot of observables are noncommutative, there are some that are. This means there exists some state $\lvert K \rangle such that it is both an eigenvector of two observables $\hat{A}, \hat{B}$. Then
the interpretation of hermitian operators "filtering" out relevant information from the wavefunction will be challenging in this special case: Suppose I have the ket $\lvert K \rangle$, does it mean that it contains information of both $a$ and $b$...? In that case we cannot really say a hermitian operator $\hat{A}$ filter out information of $a$ only from the wavefunction, as the above ket also contains information of $b$
Perhaps one way to look at it is similar to the mathematical phenomenon of some equation of motion having multiple solutions at some positions in phase space, it might just happened that under a certain basis, the state vector that corresponds to some observable $a$ happened to have the same form as $b$
And since it is true for one basis, it is true for all bases as eigenvectors of linear operators are independent of basis. So maybe, if we insist on the filtering interpretation, there are actually two $\lvert K\rangle$ that look the same but contains diffetent information of the wavefunction that corresponds to the quantum state it started with. An
alternate way to think about this issue is that most filters are not perfect (because they are linear hermitian operators, there are only so many possible states they can act on and not give the same state vector), therefore when a hermitian operator $\hat{A}$ is trying to filter information of $a$ it is more common that the filtered result will contain information of, say, $b$. This will then explain noncommutativity
But even then, there is still an interpretational issue on that: