Showing my age I was thinking of the world as it appears in ST original. I never really warmed to the later versions.
Well, hell, son, the original series was *spectacularly* American. Kirk was religious to some degree, while Picard claimed that humanity had become universally atheist (a ridiculous assertion, but one that a limo-lib like him can be readily believed to believe); Kirk was a Big Big Fan of Abe Lincoln and had memorized the Declaration of Independence; he lied to carry a sidearm and he didn't mind other people doing likewise (his yeoman even using a phaser to warm up his coffee); he was proudly masculine and loved him some hot babes of every nationality/species; he loved rules *and* he loved breaking them. He was an iconic American for the time, and I'd say for all time.
Discussion of the economy of the Federation hardly ever popped up in TOS, but many humans were shown to have private sector jobs, including numerous self employed types; there was some difficulty in a medium of exchange because even in TOS-time before replicators so much "wealth" was readily manufacturable for cheap (Kirk could not be tempted with piles of jewels because they could be stamped out in the engine room en masse). Nevertheless the profit motive was there.
Yes, the Federation was shown to have layers of bureaucracy. And in TOS it was very common to show the higher ups in bureaucracy, especially Admirals in Starfleet, as being exactly what Americans tend to think of a bureaucrats: self important blowhards who stand in the way.
The world of TOS is best explained by the wars referenced from time to time: the Eugenics Wars, Colonel Greens War, World War III. They trashed the planet, but in rebuilding, humanity finally made something good. And it appears that in those wars the United States came out reasonably well, or at least better than the rest of the world, because the Federation and Starfleet are *clearly* built along American lines. Apparently those wars were largely fought in Eurasia and, possibly, South America and Africa, with the US being only tangentially involved. And that's not unlikely: it could well be that the next major IRL war could be something like India and China taking each other out... a war costing billions of lives with the US just sitting back going "leave me the frak out of this." The nuclear winter will finally end global warming but will trash agriculture for a few years; a hundred million middle eastern/African refugees swarm into Europe and trash the place simply under their weight, while the US doesn't get the same problem because the pack ice along the equator stops the boats. When everything settles, the US is triumphant by default. And thus, we get the world of Star Trek: TOS.
Now awaiting the inevitable complaints about that not being on topic. Don't care. Trek is awesome when it doesn't suck.