I think Rally would be great too.
Agility is a lot of fun for dogs and people, but the chances of having an accident are greater in agility, there are a lot more jumps and there are obstacles the dog can fall off of, or jump off of wrong. I do not know what the incidents of dogs getting hurt are, I had 1 person, 1 time tell me that they will not do it because their dog broke its back. My guess is that, that was an extremely rare occurance. I was getting a puppy used to the equipment and we had a fall off the teeter and well, she didn't want any part of that again, but there was no injury.
Rally on the other hand, has no jumps until you reach advanced which has 1 jump, and excellent which has two. The highest jump height is 16 inches.
The neat thing about rally is that kids can compete in it. Some do very well. Some 4-H groups are doing rally. I have competed against kids in AKC shows.
Another neat thing is that positive training is encouraged. You can talk to your dog, pat your side (in novice and advanced), and do everything but touch your dog, to get it to complete the requirements. Harsh commands, or corrections will disqualify you. It is supposed to be fun for you and the dog.
Novice is done completely on lead. The lead must not be consistently tight, different judges strike points differently. For example one judge will make a mark every time they see the lead tighten. Another judge will be a little more lenient. The point is not to guide with the leash, but to use your voice and hand signals to get the dog to complet the tasks. The dog does not have to heel perfectly, but a dog that can has a solid advantage.
The shows may be the goal, but the real fun and bonding comes in the journey. A ten year old child who has the interest could train the dog himself with guidance from the parent. You can definitely train a rally competitor with a 45 minute class once a week for six or eight weeks, repeated once or twice if necessary; and about ten to fifteen minutes per day. It really does not take more than that. the child and the dog can become bored. Spend a few minutes each day working on stationary signs, and then a quick walk incorporating some of the turns and moving signs.
It also is good to train with distractions, inside, outside, outside on grass for downs, and once or twice in the rain.