I don't see anything wrong with buying a 3 yr old a pony. My daughter started riding at 18 months (before she could walk) and now she rides our friends mini bareback sometimes at being 2 yrs old.
We have two APHAs but they are only 2 1/2 so my daughter doesn't ride them. But my husband and I are planning on buying her a pony, I honestly would either prefer a POA or a Quarter Pony. Most likely the Quarter Pony as everyone I have talked to that has owned them has GREAT personalities and do very well with kids. Of course I would get something older. Right now though my daughter does ride about every other day and did you know they start lead-line barrels here at 3 yrs of age? Therefore I would love my daughter to have her OWN horse she can use for the barrels.
Oh and being 2 yrs old, she reminds ME when it is the exact time our horses and dogs should be fed, lol. That is her favorite time of the day, feeding time. She sits in the wagon while I load it up with hay, buckets of grain, and dogfood, and I just pull her to the outer fence line while I walk the food in to each animal, lol, it is sooo cute though.
I wish you luck in finding a pony though. But I agree as to what the others have said, please don't get something you know that founders, you want something to be healthy that way you know your pony/horse can also enjoy the fun your daughter will have with him/her without worrying about lameness issues in the future.