I agree that your DD is too young to get the benefit (maybe different if you were horsey) and the potential for her to be put off is high.
Even the most gentle, bombproof pony can have it a moments - especially if it gets spooked but having nervous people/children around it. What would you do it you were in the stable and the pony stood on her foot and refused to move? Or trapped her against the wall having a good old scratch? I'd know what to do because I've got horses but I would not expect a non horsey person have the confidence/knowledge to deal with it quickly and with no drama.
Much in the same way I'm no skiier and DD loves it - so if she wants to do that, she does it with a qualified person
You say most of the loaners on the yard (sharers really) are 8+ and at DDs yard it's 10+ and they have to have passed pony club badges.
You say you can book the school on the day she doesn't have a lesson - to do what? Without an instructor and the adult not being horsey I'd say that was a recipe for disaster!
You certainly won't be able to teach her and probably not be much use if something goes wrong - I mean this nicely! If you are not experienced with horses, being in charge of a 6YO and a pony is not going to be fun for anyone. And dangerous! I can't believe their insurance would cover a 6 year old riding without an instructor - so basically you'd have to pay for a lesson on both days.
You need to do a bit more digging here - would they really let her essentially be supervised around horses? With you as the adult? That would be a massive red flag for me.
It's great you are willing to learn but at this stage she needs to be in lessons or doing supervised horse/pony management- which would be extra cost.
You might end up having to pay for an extra lesson on the 2nd day. I'd say she's far too young and hold off for now.
I'm horsey and DD shared a pony at her yard for a while when she was 9/10 - she always had an instructor in the school and me in the stable (which is fine because I know what I'm doing).
On a few occasions I took her into the school - sometimes with me riding as well and it proved I was absolutely right in not becoming a riding instructor!
It is a great introduction into ponies but one and/or both of you needs to be much more experienced to get the most out of it.