Another problem is that it's certain to devalue certain characters.
If players learn that 'evil' or 'rebel' companions cause problems, what are they going to do? Simple. Avoid them. Stick to more timid and agreeable characters who follow orders and don't run their mouths. Not many people cared too much about Fist, but I can absolutely guarantee you that if bringing certain characters along seriously messed up missions and got innocent people killed, players would be demanding the companion's head on a plate. And rightfully so. And at this point, you've pretty much thrown an entire archetype of characters down the drain. You've taught players to avoid them and hate them.
Same thing with classes. On the topic of Mass Effect, suppose tech or biotic characters help out or save people on a few missions. Why would anyone bother to bring along a soldier anywhere then? Why risk it when you know biotics and techs will get the mission done? Soldiers aren't more helpful in combat. So characters like Liara and Tali get a huge boost and characters like James and Ashley get utterly thrown under the bus.
Same thing with companion pairings. Instead of letting players explore dynamics of companions who disagree with each other, you're just teaching them to avoid such pairings, and building resentment and even hatred against the characters in question for being incompetent and petty.