This is something for the barn to discuss with their insurance company, first of all.
I was a beginner rider, given a small horse on my 14th birthday, and I took on total care of her at a nearby backyard barn with no arena or trainers. I rode her down the sidewalks to the trails from day one. With no helmet. And no cellphone (they didn’t exist yet). And figured out how to retrain her bad habits on my own. Now, this was far far away in time, and there were no adults at the barn to be bothered by us. So I do think a competent rider in early teens can be left alone with a horse. And she might be competent to oversee an 8 or 9 year old.
My sister got her own horse at 11, after I’d had mine for a winter, and went riding with me up and down the sidewalks and trails. Bareback. But I think mostly girls didn’t get their own horses in our community until they were 12 or 13.
But I can also see how children loose in a big boarding barn might make everyone more nervous, than children hanging out at a small backyard facility with neighbours around, and fewer rules and other people’s unpredictable horses to get in trouble with.
That said, the freedom from adults and the chance to be totally responsible for something were hugely important to me as a teen rider. I can’t imagine growing up with today’s helicopter parents. I watched five adults (two non-horsey parents, horse owner and husband, plus coach) micromanage a competent eleven year old tacking up her lease horse for a lesson a few weeks ago, and thought if that was me, I’d have run screaming.
I don’t have a clear answer. If I was the kids, I’d feel I wanted and deserved the freedom and responsibility. If I was an adult in the barn, I’d feel some responsibility to keep an eye on the kids, and that would take away from my own enjoyment of my own barn time.
Also, what is ‘adult supervision’? Is it helicopter mum sitting on the bench in the arena watching every move, and telling the girls to cut it short cause they have to get home for dinner, even if mum knows nothing about horses and could do nothing but shriek in an emergency, and often gives the kids wrong information or suggestions? Is it an actual lesson? Does the trainer have supervised rides where she keeps an eye on things? Is it enough to have a barn worker cleaning stalls (wearing her Ipod or blathering away on her bluetooth, and oblivious to the rest of the world)? Does having another adult rider on the premises (like the OP) count as supervision even if the adult rider has no connection to the family, and doesn’t want the worry? On the other hand, isn’t it good if the kids come down at a quiet time in the barn schedule, and can putter around and enjoy their horses without getting under the feet of the dressage queens and other cranky adults?