My friend (DH's associate atty) just moved to an aging, 20 home, equine community, and the horses graze right across the fence from their yard. Her 9yo DD Thinks she has the horse bug, so she's signing her up for lessons and is going to wait 1 1/2 years to see if the daughter retains her interest. In the meantime, she has asked me for advice, and for "possible" future purchase. I told her, that if they end up buying one horse, she needs a full sized horse that ALL of the family can ride and enjoy.
Little people can ride a bigger horse and grow into that horse. We have a new thread about adults riding ponies, but all of the adults are small, under 150 pounds or so, and their mounts can handle that, so I agree with you.
Ponies change hands a LOT more often than horses do. Unless you jump in the deep end, you will all outgrow a pony UNLESS that pony is also a seasoned driving pony, too, which ponies do QUITE WELL. A horse or pony can pull up to 4x the weight that they can carry.
If the lesson pony was over 14hh, I say, maybe, but I think you'll be selling it in the future when she gets bigger and taller and heavier.
I had a 13.2hh grade pony, "Toma", that I used for lessons. He was really a full sized barrel horse with shorter legs. (the tobiano below)
I am riding the 15.1hh QH next to him (1987). "Toma" had been part of a rental string, and he had been running back to the barn, hence the sale. I retrained him and he was a faithful mount until he developed arthritis in his 20's. He lived to be 35yo and died in my barn in 2004.