Why doesn't an sp3s hybridisation exist?
The counter question that might help you rule this out is: &quot;Which 3s orbital?&quot; Note that atomic Nitrogen doesn't have a 3s orbital as N itself belongs to the 2nd period(n=2).

Also note that in hybridization, Atomic orbitals of same or almost similar energies intermix to form same number of new Hybrid orbitals of same shap and equivalent energy.
That being said, you cant expect 2s or 2p orbitals to intermix with a 3s orbital
(if it was a case. Since this case isn't possible with N)