I used to work in a private school.
It didn't select on ability (bright
/musical/sporty kids got bursaries though) and had a relatively high proportion of kids with SEND (still less than state sector). Yes, schools want kids who will make them look good to other parents, whether that's perfect GCSEs, being charming at Open Day or representing England in sport. But they also desperately need money, so if you can pay, you're in (except at some extremely high demand schools).
In terms of paying for additional help - if a child had a statement/EHCP then LA funding could be used to pay for 1-1 support, and parents were free to top that up, if they wished, e.g. a child had 10 funded hours and their parents paid for 1-1 the rest of the time. More ad hoc help wouldn't be charged for.
In general, senior management were supportive of children with SEND. They got in EPs and worked with parents and tried different strategies to help with behaviour management. However, they didn't have much knowledge re: SEND and they didn't fund much training for general teaching staff.
Some teachers were lovely and innovative and put effort into helping. I'm extremely sorry to say I heard some very unpleasant comments from other teachers in the staffroom "Why does he have to be in my class? I didn't come to the private sector because I wanted to teach the special kids" and worse.
In the state sector I've seen sensationally good schools and teachers, and also the other end of the scale.
You can't really generalise about state or private being better for kids with SEND. It depends on the kid and the SEND and the schools and the individual teachers.