They don't just select out children who are failing, they also offer scholarships to the most academically-able state school pupils, thereby further undermining state schools.
If they want to live up to their 'charitable' status perhaps private schools should take all students with EHCPs instead. Now THAT would be amazing for those kids (who would really benefit from the extra support/ smaller class sizes etc).
Anyway, regarding the OP's concerns; I agree with others that your DD will develop resilience and loads of social skills at state school. She may also do better at university if she's not been spoon fed through school but had to developmore skills for independence.
Most universities now make 'contextual' offers because they recognise its easier to get top grades in a private setting. My son had a contextual offer from Bristol and I know others do too.
Having said all that, I do understand that you are talking about your DD's experiences NOW and how they are not as rich as her friends. This is something you can address: Can she join a theatre group? Or Street dance which is amazing for self confidence. You might not be able to afford private school fees but it sounds like you might be able to enrich her life with extra activities.
Plus you can do some of the things the private schools might do with students... such as trips to castles and museums. Help her learn an instrument. She can sign up for things like Duke of Edinburgh which is very rewarding for young people.
It IS unfair that we have a two tier school system in the UK. If the private schools were abolished the upper middle class parents would focus their time and energy on improving standards at state schools.
However. Your DD has a mum who cares, is interested in her education and well being, and she sounds like she's enjoying school. Whatever you do please don't let her know how disappointed you are with her school compared to her friends! She doesn't need to know this.
Best of luck