If we avoid quantum mechanics, then spin does not exist hence it need not be considered :)
More seriously, considering spin in the basic statistical physics texts would only unnecesarily complicate the discussion - remember that these texts are designed for teaching students statistical physics rather than presenting a theoretical research in real phenomena.
A good practical reason why spin can be neglected is that in absence of magnetic field the spin states are degenerate, so the repartition of energy between them is trivial, and need not be specially considered. Moreover, many monoatomic gases have zero total spin (the ground state is often a singlet state).
If magnetic field is present however, one may have to account for spin - perhaps, people withe xpertise in astrophysics or plasma could come up with relevant examples.
It has been noted in the comments and clarified in the edited question that the point of the question is not spin of individual particles, but the angular momentum of molecules (which consists of the mechanical angular momentum of a molecules as a whole and the spins of the constituting particles). In this respect classical statistical mechanics treats atoms as point-like objects or, at best, spherically symmetric objects, so that the direction of the angular moment can still be ignored in the thermodynamic calculations.
The existing asymmetry may have to be accounted for at the level where the internal structure of atoms plays role, i.e., where the transitions between atomic energy levels are important. As an example one can consider thermodynamic state of a gas in a discharge lamp or a gas laser. Again, such cases are not considered in basic stat. physics textbooks, and rather easily treated once the basics are mastered.