Would I allow a novice to tack up a child's pony unsupervised? Yes, though I would show them how it needs to be fitted first time, then check all was ok before mounting after that. The OP says that it is checked after tacking up and OKed - not sure if that's the owner, or other yard people, but someone "trusted" is looking at it.
Would I allow a novice mother to supervise her small child bimbling around on a safe pony either on the lead rein or pottering around off it? Yes, if I was confident that the pony was a good one and the mother was sensible, and there wasn't going to be any silliness. I would worry more about a less novice combination who might be going faster / jumping higher / over-estimating their abilities. I would have half an eye out in case I saw something concerning, but that might just be me or a trusted person on the yard wandering past the school every now and then to check there's nothing silly going on.
I genuinely don't know if the pony is good, or not - I don't know that this isn't a recipe for disaster, but I don't know that it is either. And I maintain that the safest and most sensible way of dealing with it is to have a good instructor assess the situation in person, and go from there. If the pony is a good one, and these are trivial teething issues, it would be a shame for the OP to be worried into ending it, and moving onto a pony who might be far more problematic. And again, in my experience, really good kids ponies are not at all common - and nothing in the OP convinces me that this definitely isn't a good pony.
Very few 7 year olds can work horses correctly, but neither do they routinely work them very incorrectly - since 7 year olds tend to be smallish and light, and not asking horses to work against their mechanics, that's actually fine. There's more of an issue when adults interfere and try to make things "pretty", in my experience!