You said, Once an atom interacts with the screen, it no longer has a definite spin or even position.

In no event did any atom become LESS constrained in position by hitting the screen, and this applies whether or not you take individual identities or not. That's the first two lines.

The last line points out that you seem to think that the energy eigenstates of the screen and field might correspond to single atoms - but the eigenstates for an extended object will be multiparticle states of extreme complexity - and, at least within the energy regime we're talking about, a fixed number of particles.