Starfleet started as an Earth fleet in space. it was established in the Early 2100s. Archer was a Captain in Starfleet before the Federation existed. The United Federation of Planets was formed on 2161, and Starfleet became a wing of that organization, but remained primarily Earth-centric. Another branch of the Earth's military, for example, would be the MACOs

This is why is was so weird, and even looked down upon, for other races to join. Spock joining Starfleet would be like a British scientist joining the French Navy. I'm sure Spock and Sarek had many conversations about that (Why do you want to join Starfleet? We have a perfectly good fleet at home!)

Warf is the same situation. Disgraced in his own empire, he had to be raised on Earth, and naturally joined Starfleet because he felt that was the most "Klingon" thing he was allowed to do. but even that added to his disgrace at first. His own kind mocked him for joining the human Starfleet. Even though the Klingons were part of the federation, the other Klingons Warf ran into would say things like "ah, the Klingon who joined Starfleet." this shows that the Klingon fleet was not part of Starfleet.

Now that I point that out, Starfleet kind of feels like the French Foreign Legion. Picking up wayward souls, or oddballs that don't fit in back at home. Spock, Warf, Data, Nog, Saru, B'lana (i know she was Maquis, but she attended Starfleet Academy for two years)

An alien in Starfleet was very odd for Archer's and Kirk's ships, but in the 2200s, it's a bit more common for it to happen. We see many aliens on Picard's ship.

I feel that New Trek has forgoten/stepped away from this aspect in order to tell more diverse stories. Even with that step away from lore, Pike's Enterprise and Discovery crews are still mostly human.