Early in the history of model theory, people like Tarski did extensive research on Boolean algebras. We know a lot about model theory of Boolean algebras from this era. It is not even hard to show quantifier elimination of the theory of atomless Boolean algebras. I believe that Boolean algebras are tame in this sense; they should be at least much nicer than models of PA, ZFC, etc.
Later on, in post-stability model theory, the theory of atomless Boolean algebras tended to be classified in the non-tame side of theories along with PA, ZFC, and the likes. The theory has an order and an independence in a quite trivial way, for instance. This seems to me a bit puzzling considering what I said in the first paragraph.
Does the more modern model theory has something to say regarding the tameness of Boolean algebras? Or the early results about them the only nice properties about them? (Perhaps other branches of logic like descriptive set theory might explain this issue; I am interested in an answer from such a perspective, too.)