I guess I see things a bit differently. She is 8 years old and enjoying herself - I see nothing wrong with that. Riding is supposed to be fun, not work, and the dang instructor is getting paid for her time - not for your daughter's progress. IMO we need to stop putting pressure on our children and just let them be kids...
I think that this lies with the instructor not being firm enough.
I am all for them chatting but if I say "Quiet and listen' they do.
Another thing, if I was instructing and parent had to lay threats then I would not be at all happy.
Kid needs to be taken out of the arena and led from another horse for a ride, then she would concentrate more.
She should be enjoying herself but at the same time having a riding lesson is no different to having a lesson in school and if she doesn't concentrate she wont learn
Over talking can be a sign of nerves. She might seem confident but could be a good cover
Does the trainer do things with her to make the lesson more interesting - going round and round in circles is a bit mind numbing for an 8 year old?
Maybe weaving in and out of cones and maneuvering around poles laid on the ground would make her concentrate more?
I had some western lessons this winter (I'm an English rider) and none of the horses were ridden in anything but snaffles - plain or jointed - many were really experienced riders and no one rode in a shanked bit at all.
When I first bought our mare she came with a Thom thumb. I quickly changed her out of that to a shorter shanked dog bone bit. I have no problem getting her attention when she starts to veer off, but H is so small. She listens to H's cues most of the time, but when she gets a different idea in her head, plus the talking. I hate to see H have to put her entire body into stopping her.
I think I've decided to try a Billy Allen bit, as suggested to me on my other thread. After A LOT of reading, I think it sounds like it could work. A lot of people said it really helps to relax the horse. Not that Sky needs a ton of relaxing, but anything to take her mind off her baby would be nice. I'm not worried about H yanking on the reins, but it's still a mild bit just in case.