Move Sherlock Holmes from his beloved London to today's version of the city, and he'd be utterly helpless to solve even the simplest of crimes - until he had spent at least a few years learning the telltale details of modern life. Move Hercule Poirot to today, and he'd adjust far more quickly because he's a scholar of the criminal mind rather than the tools of crime. Move Dirty Harry, and he'd be equally at home (that is, not at all!) in all eras, and an efficient way to keep law and order once he learns to use the sidearm of the day.
Khan being a clever guy may mean several different things. "Space Seed" only showed him adapting to 23rd century technology in a few hours; ST:ID showed him achieving more after supposed several years of acclimatization. With just these two datapoints on the fellah, supposing he was the same guy in both cases, we can't readily assess what sort of a genius he was. But accepting that he was one should not be too difficult.
Howzabout their chunk of the galaxy is very large?
It won't be if they can build starships in a matter of weeks. Going even slightly von Neumann and having docks build further docks would allow them to field a million ships in no time flat.
Things that could plausibly keep down the numbers of starships so that "only one within reach" is still a possibility, just as observed:
1) Ships are expensive to build, the costs not being justified even by the danger posed to the Federation by the lack of naval power. But eventually, the costs would simply go down and the ships would get built by people working under minimum wages so that they might live to see tomorrow.
2) Ships are slow to build, and have limited lifespans that tie down construction resources in sustaining, rather than expansion. This might be a problem that goes away with UFP expansion, or then gets all the worse for it.
3) Ships are difficult to crew. But supposedly there are trillions of warm bodies available, and a good chunk of those might be found on those fancy arcologies we saw on Earth already. And it's not as if flying a starship is rocket science, from the looks of it.
4) Ships require a key physical resource that isn't available in any greater quantity even if one offers to pay a thousandfold price. Okuda and Sternbach sort of suggest that the warp coils might have such substances, plus their casting process is time-consuming and cannot be efficiently hastened by replicator trickery.
If the Vengeance
is quick to build, then even if this speed is due to the application of exceptional resources, Starfleet should not be a "last-minute cavalry" in general. It could distribute those resources differently into ships a tad smaller than Kirk's and rule the skies with those.