I do think that Bioware made it unnecessarily difficult to get the companion conversations. I imagine a lot of people miss out on it. The writers spent time writing it, the voice actors spent time recording dialogue, and probably the majority of players never see any of it. I think they should just unlock as you level up, with the affection level only important if you want a romance.
I really agree with this. Considering most people will probably only play through each class story once, when it is as hard as it is to make companion stories progress, so much writing, voice-acting, and animation for that matter, will only be seen by a few. Especially as it seems that there is a clear tendency for people like the same companions such as e.g. Mako and Jaesa.
A way to get around this could be as you suggest to unlock companion stories as you level up, and then simply make affection affect your dialogue options, such as e.g. romance/flirt options.
I think another way to give players more of a relationship with all companions would be to have more of those holo-gatherings, where the whole crew is present. In those cut-scenes, you both get a chance to get to know a bit about who each companion is and also about the companions' attitude towards each other.
That way, at least you could get to know e.g. Broonmark a bit without being forced to play with him.