In my experience- it is VERY different from art to art- they are NOT all alike- and VERY different from dojo to dojo. The thing about the Karate Kid style of learning vs. the Cobra Kai style is not really all that exaggerated!
My son has been taking aikido since he was about 5 1/2. Our dojo starts kids at ag 6- and even then, many are too young for it.
Many dojos have classes for younger children, but again, IMO, they are just more like exercise classes at that point and your kid is MUCH more likely to burn out early on it. My son is now 10 and is the highest ranking kid student- the kids who were big when he started are now in high school and in the adult class. He LOVES aikido, and even when we moved out of the area our dojo is in, insisted on continuing. It has been constant all this time and it is absolutely great for self-confidence, integrity, respect and all the things you said above.
but- the key, most important thing is to find not just the martial art that fits your kids best, but also the DOJO and program that fits your kid best. What we love about our aikido dojo:
1)Our sensai is fantastic in every way. She is a woman and also a college professor and is totally devoted to aikido. Aikido is not the most common martial art- it is Japanese in origin and is all about balance, controlling the situation and is not about 'kung-fu' style kicking. It is all about respect and self-control and sensai asks a lot, even of the little kids- but they LOVE her. She and the other black belts who teach command not only respect but real affection and bring out the best in the kids every week.
2)Our dojo community is very small compared to a lot of martial arts schools. We have about 20 adults total, and maybe 10 kids at any given time, ages 6-14. Because of this, it really is a community. All the parents know each other at least on sight and we know and have at least met all the adult students. It is great to really KNOW who your child will interact with. Also of course, the class sizes are SMALL. Lots of individual attention to details- not 30 little kids in a line punching and doing kicks.
There is also a great cultural emphasis. Sensai trained in Japan and aikido in general is usually pretty 'old-school'. The children are all learning a lot of Japanese in the course of the teaching and we keep a lot of traditions- such as on New Years, the entire school meets to clean the dojo for the year, and sensai grinds rice with a mortar and pestle and we share food- it is a great party- but also with a serious meaning. A lot of what we do is like that and my son really likes it. We've all learned a LOT.
3)There is a huge emphisis in aikido on teaching. My son as senior student for the kids, now leads warm-ups and is often paired up with new or younger students to help them learn. This has been GREAT for him and again, as he has been challenged with these new responsibilities, he has been proud to take them on. As you progress at any level though the belts, you are constantly refining techniques you learned earlier- so my son at age 10 is doing many of the things much older students are doing. Some adults are just starting to learn- it is all about refining what you know. One of the main requirements to earn your hakima (final stage of black belt to be an official instructor) is to conduct tests, and especially to teach. It's a really positive system.
4)It has taught my son to concentrate and sit still and pay attention. But also, I have to say that at this point, having seen my 75 pound son send a grown man tumbling away from him, one of the best things is that he is not physically intimidated. If someone- adult or another child- grabbed his arm or got in his face, I feel very confident that his internal reflexes would kick in and he would know instinctively how to twist and get away from someone, push them off balance so he could run- or how to dodge a punch or pin someone if he had to. Our sensai always tells the children that they are studying to be samurai and samurai never seek out a fight. But if fighting is put upon them, they know how to take care of themselves and get away.
Again- it is very positive,it makes me feel like my son could handle a bad situation much better than many kids his age without panicking. I have seen a little girl in our class who at first was so scared and shy she would not even come onto the mat without her father with her- now it seems like she has just gotten more and more confident and has really opened up more each week! It's really great.
5) A non-competitive atmosphere. This is a BIG issue. Aikido is NOT a sport, so although they of course have seminars and training bouts, etc. you do not compete. If you are looking for your kids to gain confidence and social skills through martial arts, you do not want a sport or particularly a dojo whose main focus is competition. There isn't anything wrong with that- but you need to know going into it what their focus is. We have been very happy keeping this non-competitive. Also, I've noticed that all the kids in class are just REALLY supportive of each other. They don't rag each other or try to show who is best, etc. It is clear- if you do it right, you get a new rank with each test. If you don't, work harder and practice more and do it again.
6) For cost- at our dojo the fee for kids is $60 per month. Each week there are kids' classes on Monday and Wednesday nights, one Saturday morning and Open Mat on Saturdays for anyone who wants some extra practice. For your $60 you can come as often as you want in a month, which works out great. Some weeks when we are very busy, we might only make it once a week. But we almost always get to Saturday mornings and do Open Mat to practice up for tests. Kid's belt tests are always $25 and sensai never has anyone test unless she is very certain they know enough to pass. There is a formal ceremony to get new rank certificates and belts usually the following week. It really means a lot to the kids.
In comparison, at the tai kwon do place my nephew was at, they charge $50 per test and then you find out on a bulletin board who passes and who failed. He failed a major test and eventually dropped out partly from the way the other kids acted about it.
Our sensai lets kids try 3 weeks of classes before committing to anything, and does not require a gi until both you and she feel sure your child is going to continue with it. Be wary of any school who asks for a lot of $$ upfront or insists that you purchase an outfit or equipment before you are sure your child wants to do it.
Do some research about what KIND of martial art best suits what you want for your kids. You will most likely find a lot more tae kwon do places as it is probably the most popular, but again, visit all the places near you, watch some children's classes and ask questions! Too many parents don't do this and end up paying a lot of money only to have their kids just drop out or get frustrated. This can be a great great experience for your kids - but you have to make sure you have the right program and dojo! Good luck!