@Oldwreck
Are you sure the independent sector is the best choice for you? Senior schools not only have to be what we think is best for the child, but also for the parent.
You mentioned before that the process is "not fair" ... this kind of thinking really doesn't go with most independent schools.
Independent schools only give prizes, honorific mentions and so on, to those that come top, pretty much like the real world, 100s apply for a job, only 1 gets it and the rest might get a thank you. Same in the independent sector, if you child won, then they get the prize or placement, if not then they don't get anything. These schools don't give prizes for just doing what you are supposed to do. It's life, nothing to do with fairness.
Equally, as a parent, you are paying premium fees, you expect the school to prepare your child. Lockdown provision was an expectation of quality as the fees didn't decrease. So when it comes to the entrance exam process, we expect our children to be ready and to do well. We expect more than one placement offer, nothing to do with how good you think your DD is or not, that is why we are paying.
Children in the independent sector are used to sit exams constantly and we know their results and where they are academically and the academic standard (which varies) accordingly. Still the 11+, etc is a process and stressful, but only as much as you allow it to be.
Apologies if this sounds a bit brutal, but it is pretty straightforward, fairness doesn't play a role as that's not why we are paying the fees we pay every term.
This is why I am asking if you are sure this is the right setting for you as a parent?
If you think it is, then best if you start preparing your DD to sit the exams.
When we moved from the maintained sector to the independent sector, we got a tutor to help our child in terms of sitting exams. Very bright academically, but has never sat exams in this way, so we did that. You can also get her books to practise the format and level of questions for the schools she is going to be sitting those exams. Lots of resources, CGP books, ISEB books, atom on line, etc.
You are a little bit behind "in the game" but you keep saying how bright your DD is, so get her the tools and get her practising, so she gets into the school that is right for her academically.
In terms of the CATS exams, these are cognitive aptitude exams, although kids can get better at them with practise, this is not recommended and they just sit these tests without any preparation.
They are not exclusively of private schools, most comprehensive school do them in Y7 if their students didn't do them in Y6 in order to stream them. Usually maintained schools do them in Y6, if not, the comprehensive school then do them as part of their admission's process or during the 1st term to stream the kids. My guess is that independent schools also offer them in the admissions process if they haven't done them.
I think that 100 to 125 is average and most kids are in this bracket. 125 to 130 is above average. 131 to 135 only 2% of the population score these (apparently that 2% live in the SW by the looks of it
it seems!) Anything from 135 to 141 says less than 1% of the population, and it is a very welcomed score in any school, alongside their present school's report, reference and doing well in the entrance exams. See what CATs score is expected in the schools you have chosen will be good too.
Hopefully she plays an instrument or excels in a sport, these come handy in the interviews.
Finally
.... as everyone has said it, co-ed places are precious and not in abundance as you have both genders going for a placements it automatically reduces the number of places for 1 gender.
Have a think and get her practising if you decide to move from one sector to the other. We moved a little while ago and haven't looked back, but we do have a very bright child with a very competitive strike and enjoys the demands that come with these schools.
Best of luck.