I think it’s difficult when your child is struggling and needs help, but school staff are stretched so thinly, and there are so many children who also need help.
It’s not simply a case of ‘a TA can sit with students who don’t want to be outside’ - schools have cut TA numbers and those left are supervising the playground.
The teachers will be using break time to pop to the loo / quick drink / prepare the classroom for the next lesson, they can’t be supervising children constantly 8am - 3pm.
As the playground is very small and noisy, could you and other parents work with the school to improve the environment for ALL students? If it’s just a big concrete space it will feel overwhelming.
For example, at my kids primary school the PTA fundraised for some trees and planting and built benches for the playground, this helped to create some shade and quieter spaces away from the football kids.
One parent also got some grant funding for the playground.
Playground toys / activities also helped focus the children’s energy, which the TAs on duty would supervise. Having the more energetic kids in one area doing catch or dancing or whatever meant the quieter kids got left in peace.
There are also parent volunteers who do reading with some children - this helped the TAs have more time with other children, those with SEN for example.
My DS found the noisy playground difficult too but he usually took a book or comic outside and would sit and read it on a bench with his snack. In that way he was able to ‘tune out’ the noise by focusing on his book. It’s a skill he still uses now and it helps him in busy places, for example at a train station.
Could you discuss with your DD what specifically she finds difficult about the playground?
I am glad that you have spoken to school now and have some solutions in place. As others have suggested try not to go in combative as you all need to work together.