Physical connection between eigenfunctions and observables?

I think I see where the confusion is (hopefully). The eigenvalue you obtain when you operate on some wave-function (assuming this function is an eigenfunction) is the value of the observable in that exact eigenstate. Every eigenstate is associated with an eigenvalue with zero variance (i.e. 10/10 times you make the measurement, you will get that eigenvalue for that state).

For example, we known the exact value for the electron in the lowest energy level of the hydrogen atom is -13.6 eV (1).

However, as these systems become more complex, we cannot obtain the exact solution for the .

(Note: I am not suggesting they do not exist, just that we do not have finite solutions to them)

So instead, we obtain all the possible eigenvalues with the probabilities that the system will have that value. If the system was in some exact state (x), it is associated with some exact eigenvalue. There isn't any variance in the eigenvalue for that exact state. The probability describes the chance that we will find that system existing in state (x) with that corresponding eigenvalue.