I've done beginner lessons in the past & will start up again this spring. I've had a variety of critters in & out of the barn over the years - all sizes, breeds, etc. Right now, I've got a teensie string of "schoolies" to use. My go-tos for the few kids I have: a devilish medium Welsh x and a Connemara x mare that's 15.1. I've also got a couple of QH mares that I use for lessons sometimes. These 4 all sit VERY different due to their conformation. The pony is the easiest to learn posting on b/c you have no choice - he "doings" the kids up & down. His size doesn't intimidate the kids b/c he's short; but he sits wider than the Connie mare; he's almost as wide as the 17hh Sea Monster. He travels w/his head up higher than the push button Connie mare so the kids actually feel safer on him - something to keep them from falling off the front, they tell me. He can be (okay, is) a turd most of the time but not to the degree of that youtube sensation pony "Ed." He's definitely not unsafe. Kids learn how to out think him. They have to RIDE him. I tend to start the tiny kids w/Turd Ferguson then bounce to the Connie mare, then back to Turd and then off and on the other QH mares as well. One of the QH mares is 15hh, narrow, travels flat kneed with decent step; the other is like a Sherman tank & so short strided - makes you wanna drink to have to ride it.
Point of all my prattle: you can't paint this issue with a broad brush stroke. EVERY horse has something to teach to the rider. Start a kid on a safe mount that responds appropriately to cues given correctly. As the kid learns the buttons, more them to something that challenges them....be it smaller or larger, it must still be safe and appropriately sized using reasonable judgment.
If our country fostered a different "pony" culture, there would be more safe & suitable & affordabley [sic - screw it, can't get it right] priced ponies. It is what it is though. Oh, and fwiw, there are TONS of QH in our area that measure a large.
I started riding when I was 8. I almost never rode ponies. Just very saintly horses ranging from 15 hands to probably almost 17 hands. I didn't actually ride any ponies that I can remember until I was a more seasoned rider, as most of the ponies had some devilish tendencies.
We don't have any ponies here on our farm - our steadiest, most trustworthy beginner horses are 16.2 hands - the sneakiest, naughtiest is 12.2 hands. Size generally has nothing to do with temperament. Yes, it is nicer usually to fall from a pony than a horse, but that shouldn't be the main thing you look for when looking for good beginner horses. Look for something safe, sane, and as close to bombproof as possible.