I've played with a lot of underaged players locally, mostly 15-17. I don't have a problem with any of them. They play hard and well with good sportsmanship and good conduct. There is an agreement with field owners that someone is responsible for the minor in some aspect, a ride home, a person who can contact the parents (if not the field owner themselves) and the parents are never too far away.
We had a kid... I think 14 get a tooth shot out. All the adults were calling no duff, sky is falling, etc... kid's only response was "I just want to go respawn" Even his parents weren't all that concerned when they were called. They showed up, they asked him if he wanted to keep playing or not, he was on the fence and looked to their guidance on what he should do. I think they ended up just going to the dentist and that was that. We saw him on the field again a few weeks later.
I have to agree though <12 is probably a bit young to be unsupervised.