1) I'm not going to argue about what's "classical thermo"
versus "classical stat mech". I'll choose another term.
2) I think we agree that there is a classical "description"
that assigns 1/2 kT per degree of freedom. It requires
us to count degrees of freedom, but doesn't tell us how
to do so.
3) When I speak of a DoF being "lost", I mean it becomes
unclassical, i.e. frozen out, i.e. no longer counted in
the aforementioned count. In the post-classical description,
the mode is still there, it's just frozen in its lowest
state.
4) The classical 1/2 kT per DoF description won't tell you
the limits of its own validity ... but there are plenty of
other theories that do have this highly-desirable property.
-- For instance, the methods that produce a power-series
expansion often give rather precise information about
the radius of convergence.
-- As a not-unrelated example, Ken Wilson's renormalization
group methods provide an internal estimate of the size
of the critical region (size along the temperature axis).
5) The experimental data on the ratio of specific heats
of gases is a handy thing to keep in one's pedagogical
tool box. It's good for answering students who ask "who
needs quantum mechanics?" It is something that can
be easily measured using 19th-century "steam engine"
techniques. The data has some remarkable regularities
that lead you to think it should be understandable, but
then it has some features that you can't possibly
understand without reference to quantum mechanics.