random effects specification

 Yes, but IIUC, this is not necessarily equivalent to fitting separate
 models with subsets of the terms/levels.

That is correct, but what is the issue? In general, I think it is best
to stay with one model. This gives you more precise estimates, because
you have more observations and fewer model parameters, for example,
you estimate only one residual variance. (Of course, there are also
situations where it is best to run separate analyses for different
parts of the data, for reasons of ease of communication, reviewer
requests, etc.).

Correct me if I'm wrong, but I
 think the estimates will differ when the design is unbalanced (e.g. some
 subjects not receiving all treatments) because a straight calculation
 from the full model assumes the coefficients have equal weight.

This is correct. Note, however, that for unbalanced designs the model
estimates correct for differences in reliability (e.g., due to
differences in the number of observations). So, for prediction, the
estimates may be better than the observed means.

Reinhold