I only watched the pilot of Person of Interest. I didn't keep watching it because the protagonist struck me as 'Jack Shephard lite'. Maybe I"ll give it another chance.
I think I gave up on it after the first couple of weeks for similar reasons, discomfort at the lead character's violence, but I gave it another chance soon thereafter and I've been a regular viewer ever since. It does have a lot of interesting ideas and is a well-made show. It's turned into a much more science-fictional show than it initially appeared, dealing with the emergence of a true artificial intelligence. Plus it's accumulated a good store of interesting recurring actors, including Paige Turco, Enrico Colantoni, Amy Acker, and Sarah Shahi.