Updated: Jun 6, 2021
A good kid's pony puts staying in synch with the rider first and foremost. If the rider canters in her body, the pony canters, but as soon as the rider's confidence wavers and she thinks about stopping, the pony should immediately stop, responding to the overall feeling in her body, even if her legs are squeezing because she is nervous. The fact that the pony is trained to put synchrony above precision keeps the rider safe and gives her confidence in her abilities.
A student will also be a better rider if she learns to ride naturally and fluidly from her seat first and to use the aids for refinement when she has the skill. Almost all children want and need this sort of intermediate pony until they are 12-14, the exception being kids in highly competitive programs. Many riders prefer this sort of ride for their whole lives.
Most ponies who compete in hunter/jumpers, Pony Club, and at Play Days are either A) not for sale, because they're great, or B) for sale, because children can't ride them comfortably. Rather than doing more ground work and giving the pony time to just get solid at a walk, the trainer probably started the pony on more advanced work like gaming and jumping to give him a job. Some trainers do this because the pony is a fun ride and they enjoy taking it to little shows and talking about how much he is just like a "little horse." Other trainers ask the pony for more advanced skills because they mistake anxiety for enthusiasm and think the pony will be happier when he is getting a little more exercise. Both scenarios lead to the same end - a pony who is light on the aids, responsive, but is not that safe.
It is so easy for a trainer to skip putting a good foundation on a pony because ponies are easy to muscle through training. Ponies with many holes in their foundation may still be able to jump a course beautifully or weave through poles like a pro, but most aren't fundamentally straight through their bodies or united in their movements. If they are straight, a rider still has to keep them between their legs and hands because the person who rode them wanted a precision ride where they had control over all the movements that the pony made.
In a good kid's pony, the default gait should be the walk. Most dressage and hunter ponies have the trot as their default gait because work is done at a trot and canter. Most gaming ponies are either loping or walking an anxious little walk where they can get into the trot or lope right when they're asked so they don't get a spanking. Both with competitive gaming and competitive jumping ponies, when the pony finally believes he is going to get to stay at a walk, he pulls the reins out of the rider's hands and mills about with his nose glued to the ground, because he is mentally and physically exhausted.
A good kid's pony should ride out on a loose rein at a walk for an hour without constantly trying to stop or getting heavy about steering. The pony should trot when the child starts trotting in her body and assume a naturally slow pace - not the pace that makes his trot look best, which is what is trained into most English ponies. A child wants a nice, slow trot because it is smoother. If the pony has been trained in the trot that makes his trot look best, you're going to have a hard time convincing him that it is better to trot slower because trotting slower can be more work. In a faster trot, the pony has momentum on his side.
A good kid's pony should trot until the rider thinks about walking, then walk - not stop, walk. Most ponies go from trotting to standing still because most trainers don't have time to ride an animal at a walk; they do all their training at a trot or canter. Contrarily, most children want to do most of their riding at a walk and are forced to do more trotting earlier on because the ponies won't let the children steer at a walk, because the pony thinks that if he gets to walk it is time to quit.
When riding at a trot, a good kid's pony should keep a steady pace, not continually speed up or break gait, like most ponies do. Rushing is a habit developed out of using impulsion to bring the hindquarters under the horse to get a nicer head set. Most kids are not coordinated enough to create and maintain a headset, but since most ponies have been trained to seek one, they continually speed up without contact.
When you take a pony who has been trained to ride well for an adult, what you see with a child rider is:
that the pony trots or stands still when he should walk
trots a fancy trot rather than one useful to a child
listens to the rider's hands and legs instead of the rider's core
continues to build speed rather than sticking to a steady pace at a trot
takes the reins out of the child's hands when he thinks he's done
Best case scenario - it will take 3 months to retrain this pony to make him suitable for your 7-10 year old child. A 12 year old could get on and still enjoy riding this pony while sorting things out over the 3 month time period, but a child younger than that will become frustrated and scared.
If you want a pony suitable for a 7-10 year old child, you need a pony who is ridden at that level and is not regularly practicing more advanced skills that are contradictory to what will be useful to your 7-10 year old rider.
Ponies are creatures of habit. Make sure you get a pony who is practicing the right things or he'll do the "wrong" things when your child is on him.