The only way you can model a dimensionless particle in QM is as a [dirac] delta function

I wasn't specifically assuming dimensionless particles. Classical atoms could be modeled particulately without being points, provided each can be picked out by a fixed position and a momentum.

In fact, you see macroscopic pointer readings. That is an important point, since Many Worlders think that the superposition disappers with macroscopic [decoherence].

Yes, this distinction is very important for BM too. For example, BM actually fails the empirical adequacy test if you treat 'spin-up' and 'spin-down' as measurable properties of particles.