The importance of learning martial arts always depends on the school and instructor. Add in kids and the selection becomes even more important because the school and instructor(s) are likely to have a meaningful imprint on their developing personalities. Training for kids is not the same as training for teens, where the training becomes more adult.
Any MMA or martial arts school should emphasize respect, discipline, camaraderie, and learning. There definitely should not be injuries or bullying. If a parent watches a number of classes, they'll get a quick impression of the school and if it has an atmosphere that fits those positive attributes. I've done decades of martial arts and the MMA school I train in now has the same traditional courtesy, respect, and careful instruction that I was accustomed to in the karate dojo.
The problem with selecting a school is that there are a lot of bad ones out there. Many MMA/martial arts schools are nothing more than glorified belt mills. The instructors don't know much and cannot teach much. The kids don't learn much. At the end of the day the parent has paid a lot of money for a multi-degree black belt and their kid can't do much more than a new beginner.
An average parent won't often see that early on, so it would help if a parent could bring along someone with martial arts experience because they can spot a belt mill within 10-15 minutes of watching a class where "advanced" kids don't look much different that the beginners except that they have colored belts.
None of the above answers the question of whether a kid should learn MMA.
The obvious answer is that martials or MMA is helpful to know in case of an emergency, but that's a load of crap when it comes to kids. Effective self defense is very difficult to learn and turning off the natural flinch-and-cower reaction when really attacked takes a lot of practice, which also includes aggressive sparring situations. It would be a very rare kid, no matter what he/she knows, who could effectively defend against a determined adult attacker. MMA or martial arts isn't magic and a kid can't overcome the laws of physics. Size and strength matter. It takes years of practice to learn how to do this stuff in real situations.
For the average kid, I think that the real world benefit of martial arts training is developing physical strength, mental strength, and self confidence. That self confidence doesn't come from feeling like you can beat the crap out of someone. It comes from working hard with repetitive practice after tedious practice, year after year, and becoming stronger and better. That feeling of hard earned accomplishment is valuable. Kids who are natural athletes have this in spades.
One one thing I'm not sure about is MMA for young kids. MMA involves grappling and striking (fists, elbows, knees, and feet). For kids, the striking has to be very controlled and there obviously shouldn't be strikes that land to the head in practice (look at all the concussion studies coming out, even in soccer from heading the ball).
If I were going to start a young kid today, I'd have him/her begin with Jiu Jitsu, the grappling part of MMA. It involves a lot of rolling on the mat with an opponent and is more than wrestling. It's very physically demanding and develops strength and speed while also learning technique. Kids learn how to get out of holds, how to escape when standing but also if held on the ground, and how to neutralize an opponent using body position and joint locks.
The Gracie family has made Brazilian Jiu Jitsu very popular in America. They were very instrumental in starting the interest in MMA. Check out some of the videos on this link to their program for kids.