Thats the thing. We think say 20,000 is high but in a 8000 light year area plus more outside exploring that really is not much!
But the other thing there is that we know that the single hero ship we so closely observe can go from one end of that area to another in a reasonably short time. Surely most of the ships of Starfleet would be capable of that, and thus the area really isn't all that big in practical terms.
Basically, even if we ignore all sorts of high outliers, both TOS and TNG vessels are capable of covering hundreds of lightyears between episodes (we hear references to specific distances every now and then, and to specific real locations slightly more often). Even if this doesn't happen quite within the two weeks between airdates, we can argue on "stardate" or "length of season" basis that a thousand lightyears per year is the absolute low end of performance. This would basically let a fleet ten thousand strong control UFP well enough to meet the onscreen evidence and the needs of drama.