Trek was, like any show, a product of its times and oftentimes reflected those times. The fact that women were apparently not allowed to be captains, for example. (Retcon that all you want with various justifications, but it's obvious that's what the writer of the time meant.)
But, OTOH, I think it's a bit unfair to suggest that Trek's non-white characters were relegated to the background out of some latent racism. All of Trek's supporting cast, white and non-white, was relegated to the background a majority of the time because, despite what some of those cast members try to say today, TOS was never an ensemble show. It was a show in the traditional format where you had a couple of stars, a guest star of the week, and the supporting cast were just that -- supporting.
At the time, characters like Sulu, Uhura, and even Scotty weren't thought of as stars, and they were not written to be stars. It definitely was not the ensemble atmosphere that would come later with TNG and DS9.
Still, despite its flaws, I think TOS did do a decent job of reflecting an integrated society. First, Spock was technicially a non-white character. He wasn't even human. But, beyond that, you did have a black woman, an Asian, a Scotsman, and a Russian making up the supporting cast. Out of a "main cast" of seven, only two -- Kirk and McCoy -- were actually white, American males.