Use your ignore function more liberally, its just great for your quality of life. Don't even stay in those end game screens nothing good can happen there. Just don't take part in the arguments and finger pointing and your life will be much better.
I don't know if there are any LoL streamers that display it well, but a Dota 2 streamers like Singsing, Merlini, Draskyl and IceIceIce have really helped me calm down when I play MOBAs. Bad teams and bad plays happen at all levels. These guys take it all in stride and just kind of laugh at the ridiculousness of it all. Draskyl less than the others, but he's cool and he mutes very liberally just so that the chat will stop complaining about his teammates' QQ.
I personally think LoL could use some in game Voice chat. It's easier than typing and can really make you feel closer to your team. I've had a lot of people add me in Dota 2 not because I'm particularly skilled, but because I take all of my mistakes and my team's mistakes in stride. I joke, I react frantically and hilariously to my own impending deaths. When someone gets angry at me, I mute them, and then just finish up the game hoping for a better next. Next time you play with friends, play on Skype, it's much more fun.
It's less complicated than it seems. I like arguing, but arguing in a game or on the internet is mostly pointless. I tend not to do it much in games because it takes a toll one everyone's performance, and you go into the next game just more drained. Kind of hypocritical though because I REALLY enjoy discussing with people who only play LoL about what Dota 2 has to offer. Its really my favorite way of procrastinating. So yea, if you feel like you can spare enough time to learn a new game, I assure you Dota 2 is a greener field provided you can play LoL at least at a 1200 ELO level.