Everyone makes mistakes, even pro players. I dont think this is an issue with internalizing the mistake, and getting better, but rather a pathological relfex that is probably done, albeit maybe subconsciously, in other aspects of there life as well.
Does the person really believe they are terrible and no friends should play with them because they are bad? I highly doubt it but maybe. Probably not though, as others pointed out this is just a defense mechanism. An annoying one at that.
I doubt they will ever get to the point where no mistakes are ever made. If so, im looking forward to there lcs debut, else they should learn to deal with it. There are alot of ways, either realizing its just a dumb game, or that making mistakes is a integral part of learning, on and on.
People handle this differently. Some rage at there teammates (worst option is to project), some turn inward (both healthy and non healthy versions- which shows great character and extreme ownership.. great traits to have if controlled), and some just laugh and go oof. If im fed and try a 3v1 and get killed, im not going to tell my friends im so bad, im going to start laughing and say oof. If they have a problem, we can talk about it.
But really, this might be a better question for a psychology forum than a lol forum. Im sure it would be the same playing cod, or wow, or basketball.