I think knitted has a point in that there is often no definitive answer as to why a parent wants private school for their.
Well f the alternative is poor then there is - but for those where the state alternative is fine, it's not so much why as why not.
We're all the sum of our experiences and environments . This is why we full our homes with things to please, comfort , stimulate us. We would survive perfectly well in a soviet style apartment.
It's why we expose our DC to certain experiences. We know it matters, even unconsciously.
Children spend seven/ eight hours a day in school ( term time). What happens there impacts on them far more than how many GCSEs they leave with. The environment and experiences if the pupils are every bit as formative as where we take then on holiday. How could they not be ?
So it's perfectly logical for a parent to want to access the most conducive environment as they perceive it.
For OP this is doubly tricky as she is giving it for one but not another and is able to compare directly.