No, they apply (or, rather, are reasonable approximations) in the specific case of objects that aren't too small, and that don't move too fast relative to each other, etc. That is, the common case, but not the general one.
We’re talking about archaeology, which is notably unconcerned with black holes or anything with a detectable de Broglie wavelength. I think that if were not going to be pedantic, and accept that we’re talking about physics as applied to Classical objects like finger bones, he’s right and there is little debate to be had. It we want to be a little pedantic we can say that QM is just about the best tested theory ever, and the rounding error left to quibble over isn’t a factor here.
If we want to be super pedantic we can point out that the “common” case on the scale of the universe is ultimately quantum-gravitational and therefore subject to physics that is not understood. Of course that majority is at distance scales we can’t access, or at energies or seem since the first second after the Big Bang. In that very very pedantic case you’re just as wrong as he is, but I think you’d rightly argue that it’s a pointless distinction in context. I’d agree with you.
[1] Standard physics isn't really up for question. Newtons Laws still apply in the general case.