Personally, I think Bakugou does care about having friends and what other people think of him. Up until middle school, being an asshole to everyone and proving his dominance through aggression was a strategy that was working for him, socially. It earned him friends, as well as admiration from his peers and praise from adults. Bakugou’s behavior is quite different from how he behaved in middle school. Right now, he’s probably incredibly frustrated by social interaction because the way he’s interacted with people all his life isn’t working anymore. (He seems to be an easily-frustrated person.) I would wager to say he’s probably also dealing with the realization that up until now he’s been praised for what he is, and not necessarily valued or loved for who he is. (That’s certainly how his teachers have treated him, and while I think the relationship between Katsuki and his parents is very open to interpretation, it seems to me to have a big impact on his character, so I would argue his parents probably did that somewhat as well, or at least didn’t counteract it.) And he’s reacting to this frustrating situation with more aggression, because that’s the habit he’s in.