Taekwondo assistant instructor here. I started studying it when I was 13. I prefer not to teach children under the age of 5 or 6 because most are not ready for it. Unless you have a very coordinated child with exceptional attention span and patience, it's just going to be an exercise in frustration for everyone. You have to know your child, however, and there are some 7- and 8-year-olds who have no business in a martial arts classroom because they just don't have the focus to participate in the class.
There are classes at a lot of larger park districts called things like "Tot Kwon Do" or toddler martial arts. I've taught graduates of these programs and often find them ready to start formal Taekwondo at 5 or 6. The difference between the tot programs and what we teach is that the tot programs are designed for younger kids--simpler, shorter, less about rigorous training and more about getting to know the body and how it works. Many contain techniques or talks about "stranger danger." It's a bit of a misnomer to call them martial arts in a way, but they can be very useful for kids who want to get into martial arts/self defense but are a bit too little to get much out of a formal class.
Bear in mind that children learn martial arts differently than adults. The vast majority of kids do not have the skills to understand the underlying physics of martial arts. Most don't even know the parts of the body all that well, and most kids are shaky on right and left well into pre-teen mode. I started studying at 13 and did well--for a kid. When I got back into it after college in my twenties, I realized just how little I really knew. It's the difference between rote learning and true understanding.
I get a lot of parents who, after 4 weeks of study, are really concerned that little Johnny hasn't "mastered" the front-snap kick. I get a lot of kids and parents who don't understand that martial arts training is cumulative, that you don't learn one thing and then move on to the next, so you forget the first thing you learned--like Western education. Given that you have some martial arts experience, you might already know this. But do try to keep good expectations--martial arts are GREAT for teaching kids confidence, coordination, balance, discipline, focus, self-control, respect for themselves and others. I've seen kids go from being complete train wrecks of coordination with no patience for themselves or anyone else to beautiful martial artists. But typically that's because the parents didn't push for them to have an adult understanding of what they were doing; they accept their kids at the developmental level they are at.
That said, I am enthusiastically a proponent of teaching martial arts to kids who are ready for it. Kids who have some semblance of an attention span and can last the hour or 1.5 hours the class lasts. Kids who have either good coordination or the patience and drive to learn to have good coordination. Kids who have some idea how to act in a classroom and can stay quiet when I'm teaching so they don't distract the other kids. Kids who can work well with a partner, practicing strikes and kicks without actually touching/hurting the other kid.
I think the best thing to do to pick a school is to keep an open mind and not stay too attached to one school or discipline. There are many martial arts out there, and there's no one art that is right for everyone. Some kids may love the complexity of Aikido or Hapkido and the idea of turning someone's force against them. Some kids may love the big movements and vigorous exercise of Taekwondo or Karate. Some kids may love to get thrown all over the place and prefer Judo. Some kids may not have much love of bare-handed fighting, but put a sword in their hands (Kendo, Koryo Gumdo, etc.) and they're like poetry. If your kid isn't taking to one martial art, try another.
If he doesn't work well with one instructor, find another school of the same art and try another teacher. A mature instructor will realize that s/he is not going to be all things to all people. A mature parent of a child in martial arts will realize the same.