What makes me like competitive games, is that I think we will never have a AI as formidable as a human opponent. So for example, game give us a set of very well thought mechanics in a "capture the flag" scenario. Humans will use the mechanics in a mix of effective/creative/fun/unpredictable way that computer AI can't simulate properly, that makes the game more fun to me. Not saying that there aren't fun bots in some games, but its very different from a human.
Not to mention that, if you don't take seriously what people say in these games, sometimes the salt is very funny as well. Problems is general toxicity. Sometimes competitiveness makes people crazy.
I don't know about co-op, the ones I played I found rather shitty tbh, that was L4D and Killing Floor 2, never had much interest in these kind of games. One can say that team based competitive games has a strong cooperative element. Insurgency is very fun in this aspect imo.