I also find that playing in a team oriented way can get others to do so as well (not playing as a team is the best way to get frustrated.) Throw armor on a few people as Symmetra, bring a wiped team back to the objective quick with Luccio, or even slap down some well-placed healing stations as Soldier 76 and most people will start acting in a more team-oriented way.
The game is designed to put very little emphasis on winning, with a relatively low XP reward and no real post-game stats screen for teammates, so I focus on playing the objective, playing my class, and letting the chips fall where they may. I don't really get frustrated with the game even when losing badly because it's just an opportunity to hero switch and try a new tactic. It's all based on how I feel at that specific moment in time. I don't want to sit there and wait for 15 minutes for so-and-so to show up. I don't have a certain number of games I want to play. Much like Jeff, I tend to jump on randomly, get a few games in, and take a break. Playing with a premade group is pretty much also out of the question for me. Thank god the matches are so short though otherwise I would have already quit. But one-sided matches aren't fun at all, especially when you're on the receiving end.
Shit happens and there can only be one winner. It's like "I can't find a way to help my team win because either they don't give a fuck or we're just getting outplayed and there isn't shit we can do." It can still be tons of fun losing games where there's a good back and forth and it feels as if your entire team contributes. It's that feeling of powerlessness that drives me nuts. There's a lot of shit that can happen in this game that I personally find frustrating and it's why I too play in short bursts.