Metagame-wise, I wouldn't have a problem with a die roll, but my game is one of those that you describe, where people are more invested in their characters, and doing things that would be against their character's desires (even in metagame, like assuming they come to a reasoned decision, but for the sake of expediency we roll a dice) are generally seen as second-best solution. Players want to roleplay it out, players get to roleplay it out!
Having said that, in the session last night, they took my hint, and were much more well-ordered, and in my opinion, had a lot more fun acting like a team than like a troupe of mass-debaters.