You can play through the story part of the game with any of the characters if you gear them well. You don't have to stick with the obvious choices. Like any Bioware game, you get to know the characters you spend time with. That being said, if you're crafting, which you should be for numerous reasons, you can get enough companion gifts to raise the other characters to the point that you can get to know them as well. Getting to know the companions is as much a part of the game as gearing yourself, doing your class quests, or doing flashpoints, and requires the same attention.
This is pretty much what I was going to say. If getting to know all of your companions is important to you, it is something you have to actively work on. 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 realized the problem early on, so my solution was to level up Treasure Hunting and Diplomacy on my first character, and just constantly churn out companion gifts. It takes loads of them to get all companion affection levels to max to unlock all of their dialogue scenes, but in the end I think it is worth it so that I can see them as real characters and not just strangers that hang around on my ship.