In the Uk we have a diverse selection of ponies and most will carry a lightweight adult. Heck many will carry a heavier adult for a while!
Welsh ponies (Section A& Bs) use to be used by farmers on the Welsh hills to carry not only the farmer but also a couple of bales of hay (one on the farmers back and the other balanced across the ponies neck) Ditto with the Dartmoor's and Exmoor's and Shetlands.
All the 'natives were work horses, Fells, Dales, Highlands, New Forest and Connemara's had a purpose in life.
It is easier to start a small pony with a rider as they very rarely ever object (I always say it is not the weight but the shadow they see above them that causes problems) Ponies are use to people being above them.
I start them the same as I would a horse, then I will use a better child rider to start riding them and I am never 'nice' to ponies in that I will have kids vault onto them, do round the world, leap frog onto them and generally get them use to all sorts of silly things a child will do with them.
Many times I have had 'problem' ponies sent to me to 'sort out'. They are only problematic because they have learned what to do to either ditch a rider or avoid work.
One such pony I bought for £50. He was fantastic looking, bred for showing, moved like a horse and was obviously as sharp as a razor.
The problem lay with the canter. One stride and his head went down and he would send a rider flying. Brother could that little beggar buck!
My niece rode him, she was good for her age but, he would send her flying every time he cantered. One day he broke into a canter on the road and she was pretty shaken. I was so mad I got off my horse, stuck niece on him and got on the pony (13.1) I bear him straight into a canter. He found it hard to buck with my weight, cantered him up the hill. Along the road, past the stables and up a very steep hill and then continued to the top of the hill - a distance of about a mile. He lost all thought of bucking by the time he got to the top,
I actually rode him for about four days, my feet were below his knees and he felt my weight but, I really didn't care. He was no good as a child's pony if he thought it good sport to drop them every time they cantered.
I rode him up a very steep hill - he wanted to stop half way up but I kept him going. I rode him down the hill and then put my niece on him. He never ever bucked with her or anyone again.
At 5'9" and 140 lbs (a long time ago) I would get on an 11.2 that was being naughty with a child rider. They soon learned that bad behaviour meant work and hard work at that.
As long as they respect the adult they will behave with the children!
Naughty ponies usually go very well when the child has outgrown them and has the strength to make them behave!