As someone who has been involved in various martial arts for over 30 years (i was 5 when i started), i can say that there is NO WAY any responsible club would allow kids of this age to engage in full contact bouts. I teach Kempo Ju-jitsu and Arnis to adults, and karate to children and there are many ways for children to train in martial arts without risk (semi contact for example). At that age, bones are not developed and this kind of thing can (and does on a regular basis) cause deformities and permanent damage. I don't even allow juniors to use the punchbags for that reason.
I also don't recall anybody saying that boxing is ok for this age group. In my opinion, that is no better. Hand and wrist deformities and arthritis in a child are not acceptable risks. And anyone who thinks "cage fighting" or "MMA" is the same as wrestling is deluded. In sport wrestling, it's all about grappling, takedowns and holds - no strikes allowed (and in my opinion this still shouldn't be practiced by minors for the reasons outlined above), and TV wrestling is a pre-arranged show which is again only performed by adults.
The football argument has also been offered out of context. Deliberate fouling and faking injury are mainly the habits of adult footballers, not children. Any coach worth his salt cracks down on that kind of stuff.