ETA: I read through the responses, and think everyone had good things to add. Ultimately you have to be comfortable as a parent with the level or risk you want to assume. This was a good article in Time - http://time.com/4948320/football-brain-damage-consussions... if it helps.
Our kids played sports - primarily hockey, soccer and basketball. As I mentioned, we've seen kids injured. The thing with football is - it's the amount of head rattling (thumps) that goes on when they are young, and they have proven that head thumps are what cause the damage - it doesn't have to be a conclusion. Granted, they are studying ex pro athletes in a lot of these studies, but they're extending it now to kids who play at school levels. I think it's best to just keep informed, keep perspective, and it's good to talk to your kid's doctor and get their take on it. They have to stay informed - they read the journals. They see the concussions and injuries firsthand.
I agree you can't bubble wrap kids and a lot of times my kids have had bumps, cuts, and hospital visits just by flukes. But the thumping and jostling of brains - from tackles and such ... it's like competitive hockey (slamming into boards) - we said no. That's just a personal decision.
-----------------------------------
I have seen the shows (documentaries, 60 minutes, PBS shows, etc.) on it about brain damage.
My husband has had 2 concussions. He has migraines regularly and it has also altered his personality somewhat (after the 2nd one). That is what concerns me the most. The doctors have said if he is to recover somewhat, he'd have to rest in a dark, quiet room for at least a month. There's no way we can make that happen at present so he just suffers through it.
We've known kids to have concussions in hockey - and that's just at the recreational level. Unfortunately, it does seem like it's the same kids over and over - typically, the smaller ones. They also have had the broken bones (the kids we know).
It can happen to anyone - a friend of mine slipped on ice, had a concussion and now has bad vertigo.
Personally, I think if you can avoid a situation where you know there's a high chance, it makes sense to not participate. However ... I haven't been in your spot ... I know it's hard to talk a kid out of something they're mad about.