But, for a system of two quantum particles which interact according to a central potential, you can express this using two quantum non-interacting particles one of which corresponds to the center of mass of the two, and the other of which corresponds to the relative position, I think?

And, like, there is still uncertainty about the position of the "center of mass" pretend particle, as well as for the position of the "displacement" pretend particle.

(the operators describing these pretend particles can be constructed in terms of the operators describing the actual particles, and visa versa.)

I don't know for sure if this works for many electrons around a nucleus, but I think it is rather likely that it should work as well.

Main thing that seems unclear to me is what the mass of the pretend particles would be in the many electrons case. Oh, also, presumably the different pretend particles would be interacting in this case (though probably just the ones that don't correspond to the center of mass interacting with each-other, not interacting with the one that does represent the center of mass?)

So, I'm not convinced of the "nuclei have to be treated as frozen, otherwise electrons don't have a reference point" claim.