I'd add to what Ben says the observation that people have found a number of different module categories valuable for different purposes within this same Lie algebra context. (And some Lie theory people don't really find category $\mathcal{O}$ to be all that important in their own work.) Even two people close to the original construction, Joseph Bernstein and Sergei Gelfand, found it more useful to broaden the study to categories satisfying somewhat different finiteness conditions in their further work on projective functors.
Much of the original motivation for category $\mathcal{O}$ came from a rethinking of the classical finite dimensional theory combined with an attempt to understand better the problems raised by Verma's thesis and later work by Jantzen. Here is where the translation functors really come into their own, along with the refined use of central characters and blocks. But the "correct" module category to study depends on which problems are being studied. The category of all modules for a universal enveloping algebra is definitely too big for practical purposes, but within it there are many attractive subcategories.