I was thinking about what would the different Star Trek shows look like if depicting queer people was just a-okay at the time, without making drastic changes and without taking many of my ships into account (I am showing so much restraint here). Here are my thoughts (no TNG, I haven't seen that yet):
TOS: Kirk is definitely having romantic flings with aliens of all genders, Sulu is openly gay, we have sexy clueless crewmen of all genders, BUT crucially despite all this Kirk and Spock's whole thing is exactly the same, Bones frequently grumbles about how they should finally fuck because the tension drives him crazy, but they never do (until the movies)
DS9: Jadzia is even more free in terms of sexuality but also gender expression and uses a different set of pronouns every couple of episodes or with different people; Rom is a trans woman, which offers interesting plot lines because of Ferengi gender politics; Odo is even more agender and asexual than before, but we also have at least one more character (in addition to Dax) who is more fluid (no pun intended) despite presenting quite stereotypically as a man or a woman (I propose Leeta for this); Garak and Bashir are definitely gay and will get together after a few seasons of the exact same insane behaviour that was already present in the real show; Kira is bi or pan but exclusively dates the most boring people in the quadrant; Gul Dukat is into Sisko sexually but nothing is changed from the original
VOY: Janeway is a lesbian with a bland female fiancée she leaves on Earth and all of her (very few) romance subplots are with women, however she still has something going on with Chakotay, they never get together and everything between them (the same as in the show proper) is therefore even more confusing and infuriating for the viewer; Seven is either asexual and there is a whole episode about her discovering this about herself or she is gay and ends up with Janeway in the finale, which is still very unsatisfying for most viewers somehow; Tom Paris is trans probably; the Doctor experiments with his sexuality and gender subroutines every couple of episodes just for shits and giggles
ENT: lot of heavy lifting needed here... ok, ok... Trip and Reed fuck, but don't end up together; Phlox introduces his wives and husbands during the show; Archer is inexplicably repressed af, despite being 100% straight... honestly I am struggling, somebody help