Again, sure, I can see it, kind of. Except that you've got things like the Klingons, who have the resources of hundreds of worlds - including numerous vassal races - for a thousand years before the humans first moon shot.
Yet, Klingon ships are significantly less advanced than human ships. Note, sure, the Enterprise is a Federation ship, but, it was built by humans. None of the other federation races had empires even remotely the size of what the Klingons controlled.
So on and so forth. It's so ingrained into the basis of the show that it's almost never questioned - of course humans are just as advanced as everyone else. We're obviously smarter, more creative, and better than everyone else, so, of course we're on the top of the heap.
It's an attitude that you find in genre fiction all the time. It's such a part of genre fiction that it doesn't get questioned, it's just assumed to be true.
It's an assumption that's crept in over the decades.
TOS had a stronger implication that humans reached space by themselves and had a bit of a rocky start (war with the Romulans). The humans were able to cobble together the Federation from the individual worlds that they found had spaceflight, but were working alone. The Klingons were another expansionistic culture with a growing border with the Federation. More a Soviet stand-in than a Mongol horde.
TNG introduced the origins of the Kahless about 1500 years prior. I think TNG was also the point where it was stated/implied Shakespeare was a shipwrecked Klingon, but that might have been DS9.
STE introduced the Klingons as the major power in space (belligerent but passive/seeking good fights) and having D6/D7 style craft already.