I wouldn't pay for lessons for a three year old. The other things at the barn, the crafts, movies, etc.. those sound like important social exercises, so I wouldn't quit those.
Just the fact that the trainer seems a little blase' and passes your granddaughter off to people who let the pony bolt... worries me. Firstly, that trainer should be WAY more attentive than that. [You aren't paying her to water a houseplant. This is your granddaughter!] And secondly, no lesson horse/pony should behave that way.
It sounds like what you guys do with your Standie is perfect. She gets to brush the horsey and make friends, and learns to move with the horse's body. Eventually you can give her reins, while still leading her, and ask her to steer a little bit. As long as you keep lessons short, fun, and easy, I think what you're doing is perfect for a child her age.
We drink to our youth, to the days come and gone; for the age of oppression is now nearly done.