The barn isn’t going to change. Even if you have a come-to-Jesus meeting with coffee and doughnuts with the instructor, maybe, maybe your daughter’s lesson will start on time once. (Although I wouldn’t give coffee and doughnuts to an instructor who wasn’t giving me her time and attention.)
But in a barn where there isn’t enough help and a low commitment on the instructor’s level, it’s not going to change week to week.
So many lesson barns are like this. The kids pick up skills from just being on a horse, not because the instructor is doing or saying anything but telling them to jam their heels down or having them perch in two point. The instructor probably doesn’t really know how to teach (at least not beginners or small children) or is totally burnt out is incapable of seeing what kids might need to progress (which requires observing kids). Or maybe there’s just too many students and not enough help. That’s not your problem.
I really wouldn’t pay money to have a child ride around at a walk for less than the allotted lesson time, perhaps picking up bad habits, while the instructor is on the cellphone.
I agree, ride at home, get an instructor for the kid at home. Take advantage of the situation you have (a pony). Give the good instructor your hard-earned money and buy her coffee and doughnuts instead, just for being awesome. If your child wants social time with others, join another activity (with a competent instructor).
Also, if the instructor has 5 students in every lesson, and 4 have parents who are absolutely satisfied with nearly no instruction and things starting late, it’s actually easier for her to get rid of you (sorry OP) and find another student with parents who are happy with chaotic lessons and she can just happily post on Facebook the entire time. I personally have never been at a barn that changed ANYTHING because someone complained. There was usually a reason things were done the way they were, even if it wasn’t “correct” or “ethical.”