I'm not sure about this "manageable Academy" thing. For all appearances, the place is the size of San Francisco. And we never heard of classmates, really, of heroes knowing each other from the Academy days already. Data may have graduated from a class of 78, possibly Mrs Robinson's, but there's no telling how many classes are running in parallel. Attendance could well be in the millions, much as in the bigger universities of Earth today.
As for there being 12 starships in TOS, I'd argue the exact opposite: when Kirk says there are "only" a dozen like his, this establishes that it is rare for any particular starship class to be that small, and most have dozens upon dozens of hulls.
Why would manpower be a problem for the UFP? In Britain, with a total population of 60 million or so, fewer than 60,000 are needed to run the Royal Navy show - one in a thousand or so is already overkill. The UFP in the 24th century has a population in the trillions, and it just doesn't strike me as plausible that billions would be needed to operate mere thousands of ships and installations. Millions, perhaps - but that's already a thousandfold improvement over the Royal Navy woes! If the Royal Navy had to rely on skilled volunteer hobbyists to run everything, it would not be particularly hard pressed to find sixty of those...
Timo Saloniemi