You're definitely taking this picture too seriously. Hybridization of atomic orbitals is just an approximation based on an independent particle model created by Pauling to rationalize some structural trends in chemistry with Quantum Mechanics. We can only assign orbitals unambiguously for one-electron systems, though of course independent particle models were widely employed and still are to explain reactivity trends, etc.
Anyway, if you prepare an electron in an sp3 state, which is an equally weighted linear combination of the px, py pz and s orbitals, the probability to find and electron at a certain angular momentum might be calculated by taking the square of the projection of this angular momentum on the wave function, as QM tells us to do.
Don't worry too much if you find weak spots in general theories developed to explain molecules, reactivity, etc, since they're highly approximated in most cases (unless you do an expensive computation for a system, but then it is not general anymore), and the guys that developed those such as Pauling already knew that. As you know these highly approximated theories (e.g., Molecular Orbital and Valence Bond theory) were and still are very successful, even if they're not very rigorous.
It gets really hard to be rigorous in chemistry beyond a certain point...
Of course the basic formalism of QM would predict properties of this hypothetical electronic state, I did not say anything that would imply the opposite.
Still, to think about bonding in terms of sp3 hybrid orbitals such as one does in general chemistry is really a very crude picture, unless you're dealing with molecules that have very special properties such as Td symmetry (methane, for example), and still in those cases if you perform a calculation by using VB or MO (Hartree-Fock) theory (and it does not even need to be with a computer, since symmetry is going to make life very easy here) with only the SP3 orbitals of carbon and S of hydrogen in your set of basis functions, you will see a very big quantitative error in predicted ionization energies when you compare to photoelectron spectroscopy measurements of ionization energies. Another good test would be to perform high level quantum chemistry calculation of methane using a software and employing two different basis sets, one containing only s and p functions centered on carbon and s functions centered on hydrogen atoms, and another in which the basis set has functions of several angular momenta centered on each atom. If you compare your results with experiments you will see that the first method will have a much lower accuracy compared to the second one. S and P functions might still be the ones that contribute mostly to the bonding molecular orbitals in the different orbital configurations (set of occupied molecular orbitals) that contribute to a good description of methane, but we can only say that for sure for very special cases. When you go to molecules with very distinct geometries, it is vital to have d functions centered on carbon, for example, in order to determine even qualitatively right chemistry trends.
Quantum chemistry semi-empirical methods widely used in the past often had this problem of only employing valence basis functions centered on an atom in molecule calculations, not giving enough flexibility for the description of molecular geometries, and therefore providing bad results whenever the geometry deviated too much from what the qualitative models (such as the hybridization model) predicted.
In fact, I'd really be surprised (but not too much) if any modern paper that discusses chemical bonding, or that qualitatively discusses bonding orbitals to explain computed trends, do that by employing hybridization theory arguments. Those that I have seen almost always do that by looking at molecular orbitals or some refined valence bond treatments.
Therefore, although hybridization is a deep and important topic that every chemist should dominate, in the way it was developed by Pauling it has severe limitations that people are, or should be, aware of.