As for the title of this thread. No. What it is, even if it was just the main quest giver, is that it would be limited to the main quest giver. This doesn't mean "empire is gay and republic isnt"
First off, your character who's gay would already make that statement untrue.
Second, it just means that would be the only NPC currently with the option. In fact, not having the option currently doesn't mean there isn't gays in star wars, what it means is, you dont have the flirt option.