I would soundly endorse a game mode, possibly even just a version of ARAM wherein there are NO report buttons after the match.
Only a simple plus. And you'll accumulate a (plus percent) based on the ratio of matches where you get plussed for whatever reason.
Could be teamwork, friendly whatever but in that mode it's just a plus.
If after a while you are severely NON plussed then you will only be matched with other people that no one likes.
Think of it as a mode with restricted utility where the important parts are a) horsing around and b) blowing off steam.
Do not say that custom games already have this covered. I want a place to put the players who have no one willing to play with them.