Romances are a touchy area for me. To be entirely honest, I tend to enjoy them, but half of the nerdy pleasure I derive has to do with what non-romanceables lack. In general, characters that aren't being romanced have fewer dialogues, and aren't explored to the same depth. While you may become familiar with their backstory and their personality, the greatest degree of reactivity is found in romanceables. They'll react to more stimuli and be more aware of your comings and goings. You influence them to a greater degree, and they feel the most grounded in the world. If we had characters that had as much of their backstory explored, as much of their personality revealed, as much of their opinion to a current event shown... I think my like of romances would dwindle. It would still be present because it's an interesting angle to explore in roleplaying, but honestly, I have a strong love of bromance.
Having a character in game who you can not only become friends with, but very good friends, would be amazing. Someone whose dialogue you can play off of, someone who you know will be there for you, even if you choose not to romance them, that would be something else. Sure, we had characters like Imoen and Mission, but they never hit the bromance mark. Both felt more like a kid-sister you constantly had to rescue from danger (Imoen especially). She felt more like someone who leaned on you for everything. You were her rock, but when it came time, it never seemed like she was yours.
I'd also like to see a sort of rivalry bromance or relationship develop. Someone who respects the hell out of you, but might not necessarily like you. Maybe someone you can have a battle of wits with? I usually find that opposing characters are some of my favorite, and a personal and interesting relationship with an adversary, or grudging ally could be an interesting route to explore. Which is another thing that I've noticed. Aside from romance being the preferred/only method to become thoroughly acquainted with a character, friendship is the other. Can a character not express themselves in detail while disagreeing with us? Granted, generally one is more inclined to talk with someone they're friendly with, but given their presence in the party, I generally assume they aren't completely vitriolic in their feelings for the PC. And disagreeing with you can consist of more than just whining or saying "no".
That being said, if bromances or rivalmances or sibling-like relationships were implemented, I would be a happy camper. Romances are all well and good and something I'd still explore with characters for whom it makes sense, but they wouldn't be the only way to sate player curiosity.