I think you're on your way to coming up with the idea of an expected value. The additional step of reasoning to get there is to think about what number we'd use to describe an outcome that had a 50% probability of going "forwards" and a 50% probability of going "backwards." If 0 makes sense as an answer to that, you're doing a probability-weighted average of outcomes. Quantum mechanics is full of expected values, and you can interpret wavefunctions just fine with them.
In the QM formalism, we come up with these operators that can be used to compute expected values from wavefunctions, by doing a calculation that, if it were involving vectors, would be written like E[A] = x^T* Ax, where x is the wavefunction, x^T* is its transpose and conjugate, and A is a matrix designed to pull out an expected value when used in that way. The demand that the results of this calculation be real for every possible wavefunction give us the property that A is a hermitian (A = the transpose and complex conjugate of A, it's like being a symmetric matrix), and from there we know that A can always be diagonalized. If A can always be diagonalized, we can always write x in a basis that diagonalizes it, in which case x^T* Ax becomes something that just conjugate-squares the magnitude in front of each eigenvector and multiplies it by something on the diagonal of the now-diagonalized A. If you go back to the original definition of expected values as a probability-weighted average, the conjugate-squared terms are playing the role of probabilities and the eigenvalues on the diagonal of the matrix are playing the role of outcomes.
In the QM formalism, we come up with these operators that can be used to compute expected values from wavefunctions, by doing a calculation that, if it were involving vectors, would be written like E[A] = x^T* Ax, where x is the wavefunction, x^T* is its transpose and conjugate, and A is a matrix designed to pull out an expected value when used in that way. The demand that the results of this calculation be real for every possible wavefunction give us the property that A is a hermitian (A = the transpose and complex conjugate of A, it's like being a symmetric matrix), and from there we know that A can always be diagonalized. If A can always be diagonalized, we can always write x in a basis that diagonalizes it, in which case x^T* Ax becomes something that just conjugate-squares the magnitude in front of each eigenvector and multiplies it by something on the diagonal of the now-diagonalized A. If you go back to the original definition of expected values as a probability-weighted average, the conjugate-squared terms are playing the role of probabilities and the eigenvalues on the diagonal of the matrix are playing the role of outcomes.