I think the reason to cooperate is not to get the best personal outcome, but because you care about the other person.

I just want to make it clear that by saying this, you're changing the setting of the prisoners' dilemma, so you shouldn't even call it a prisoners' dilemma anymore. The prisoners' dilemma is defined so that you get more utility by defecting; if you say you care about your opponent's utility enough to cooperate, it means you don't get more utility by defecting, since cooperation gives you utility. Therefore, all you're saying is that you can never be in a true prisoners' dilemma game; you're NOT saying that in a true PD, it's correct to cooperate (again, by definition, it isn't).

The most likely reason people are evolutionarily predisposed to cooperate in real-life PDs is that almost all real-life PDs are repeated games and not one-shot. Repeated prisoners' dilemmas are completely different beasts, and it can definitely be correct to cooperate in them.