Harry didn't play DnD or MtG though. He lived in a magical world, and while some of those magical things occasionally caught his attention, he wasn't really deeply interested in any of it. His main concerns were social: casting unsubstantiated aspersions on Snape and Malfoy and occasionally being correct. He didn't care about any of his classes. His only real love was Quidditch, which he was naturally talented at, making him, if anything, a jock.
Rethinking through the series, which I have re-read within the past year, the only truly nerdy moments he has all involve girls; and then only really at the beginning of his involvement with girls. Very quickly, he begins to have quite mature thoughts regarding his relationships, even if he doesn't always make the right choices.
Harry Potter addressed this when Harry learned that his dad was Snape's childhood bully and cuckold.
Tina Fey also did a good exploration of this idea in 30 Rock S3E5 "Reunion" when Liz Lemon has to face the reality that she wasn't the nerdy victim but the bully. It's a recurring theme in media because no one ever thinks they're the bad guy. It's especially hard to tell when you're young and haven't learned how to be a human yet.
All that said, I don't think Harry was a bully. Malfoy was pretty unambiguously antagonistic. Snape was more antihero, but did make a point to antagonize Harry when he could. Harry mistook the antagonization as evidence that Snape was up to bad dealings because "I'm the good guy and my enemies are bad", but even this gets flipped on its head throughout the series. Malfoy doesn't become humanized until they show his family life in detail and even then he only redeems himself in the home stretch.
Rethinking through the series, which I have re-read within the past year, the only truly nerdy moments he has all involve girls; and then only really at the beginning of his involvement with girls. Very quickly, he begins to have quite mature thoughts regarding his relationships, even if he doesn't always make the right choices.