Great question, OP. And one I often wonder myself.
I have one child at an independent specialist provision (as pp have said, paid by LA), and 2 children at mainstream independent schools because they would not cope in mainstream state provisions.
All 3 would be utterly let down, their education compromised and their mental health severely impacted if they were forced to attend the equivalent state mainstream provisions.
My middle child has severe anxiety (along with other disabilities), and some days even the thought of going into her small, totally friendly, completely understanding school (16 children in her form and teaching group) is beyond her. There is no way she would survive in our local (very well thought of, sought after, etc) secondary, where the intake each year is close to 300 pupils - that’s over half dc’s entire school! And that’s quite apart from the fact that with numbers such as those, the flexibility that is sometimes required to help her get through the day is just not possible.
My eldest has only ever attended independent specialist placements (following a disastrous time at mainstream preschool). She has learned so much from her (admittedly eyewateringly expensive) excellent school, and it is her best chance at even a semi-independent life. Without the ridiculous tribunal we were forced to go through to secure this placement, she would have had a Statement (as was) offering 7 hours help per week from a non-skilled TA. She currently has full time highly trained 1:1. That’s the kind of gulf we are talking about, and is ultimately about providing her with a decent quality of life.
There is so much crap talked about just closing private schools, and the people who arguably benefit most from attending them are usually totally overlooked or ignored (like most areas of life, tbh)