To directly respond to your first question: yes, "density" is quite common in middle-to-top tier grad programs in math in the U.S., once one gets beyond the "required/core" courses. There is a lot of backstory for almost all subjects, and faculty want to catch up to somewhere close to current events. It is also common that one's conception of an incisive, efficient, insightful course is not mirrored accurately in existing textbooks, so one writes notes to match. (This is better than "in the old days", pre-TeX, when most often there were no published notes of any kind, not even hand-written, so note-taking in class was critical.)
About "abstract" functional analysis in particular, yes, I think it is fair to say that the style of many of the standard texts (and of practitioners) has drifted in a certain direction, including a sort of "density". The hilarious and not-really-helpful opposite, from an older time, was the Dunford-Schwartz 3-volume extravaganza, which treated nearly every bit of mathematics as though readers would not have heard of it. Determinants, polynomials, etc. Naturally, this low-density high-volume approach created its own problems, not the least of which was locating the key examples that were not part of "general" mathematics, but specific to the functional-analytic ideas.
The subsequent decades' monographs or texts on "topological vector spaces" have always struck me as peculiarly unhelpful in making connections to other parts of mathematics, though "maybe it's just me". I cannot keep all the possible adjectives straight, and I do not see the point of many of them, nor are examples clear to me. Again, maybe it's just my own limited interest, since I care about "functional analysis" for "applications" (to automorphic forms and related matters) rather than to itself, or to "applications outside mathematics".
A point that was not clear to me for a long time is that the "more exotic" types of topological vector spaces (and families thereof, as in Sobolev-space theory) are not merely "handy", but essential in making many innocent-seeming discussions legitimate. So, not merely Frechet spaces, but LF-spaces (strict inductive limits of Frechet) already appearing as the correct topology on $C^o_c(\mathbb R)$, for example. The notion of quasi-completeness really is necessary, since LF-spaces are never "complete" (unless they're secretly Frechet), but are always quasi-complete. And quasi-completeness is enough for many essential things to still work, like the Gelfand-Pettis (or Bochner) ideas about vector-valued integrals, and Grothendieck's results about holomorphic vector-valued functions, and the strong operator topology (weaker than the Banach-space uniform norm topology on operators on a Hilbert space). Holomorphic distribution-valued functions, and so on. (My on-line functional analysis notes certainly go in such directions.)
In short, examples are critical, both to illustrate adjectives and theorems, and to see the necessity of introducing various seemingly-exotic types of topological vector spaces.
At the same time, I think building up a resistance to random counter-example demands is good, much as with point-set topology. Some counter-examples are significant, others aren't. Much time can be killed, not terribly profitably, by trying to illustrate every possible logical distinction. Don't do it.