I think you have the wrong concern. Private schools can vary as much as state, though they often have an edge in terms of class size, teaching time, and selected pupils. And at A level it is more than possible to have a weak subject teacher in either environment - for example there is a national shortage of maths and STEM teachers and both sectors can be affected.
You should look at recent University destinations and A level performance in the subjects your daughter will be taking, after factoring in the selectiveness of the pupil intake, to see if they reflect your daughter's aspirations. Hopefully as part of the selection process she woud meet the subject teachers. Look also at the broader opportunities. Sixth form can be a key time to get involved in first team sport, drama or music and a chance to lead. Absolutely observational, but experience in stepping forward, perhaps presenting to a school society or entering an essay competition, stands you in good stead at University where self starting is important.
Others are right. Contextual offers are usually only a grade, and can include private schools Bristol University's list here But observationally top universities like Cambridge and Imperial are giving quite a range of offers for STEM subjects based presumably on interview performance and educational background. But the observation is of strong pupils from top performing schools where A*s are the norm, and so in line with what the better pupils should be achieving.
As a previous poster said, the first term may be tough. It depends on subject. I don't think maths/STEM is so hard as long as you have the ability and decent GCSE results. DD switched to a more academic sixth form and has loved the faster pace. Those that struggle have probably chosen the wrong subjects or are not working hard enough. In contrast her peers taking languages and classics noted an immediate step up and found the first term tough, as the new school taught well beyond the test at GCSE. The best advice would be to get a vocabularly list and learn it over the summer. Humanities also seem to have been different with smaller class sizes allowing for more challenge and debate. Useful skills for an Oxbridge candidate to master. But this is one school. There could be other private schools that get their results by spoonfeeding and teaching to the test.
So don't worry about University entry. Instead look carefully at the quality and breath of education provided.