Would it be accurate to say that the "Vibration" or nuclear-spin flipping doesn't emit energy or is neutral in the thermodynamic sense?
That said, some aspects of thermodynamics are counter-intuitive (well, actually most of it is). Vibrations are present even at 0 K because of the uncertainty principle, and that’s true for spin waves as well as actual vibrations. It does not mean that any energy is created. The difference with time crystals is that usually these vibrations don’t involve any symmetry breaking.
Time crystals are consistent with standard thermodynamics.
Are you sure that's true? I can't remember the exact details but recall that being a common misinterpretation of the uncertainty principle