Private Education for 'average' kids

Yes, some secondary private schools are for the clever (even very clever) kids. But there are plenty of private secondary schools for the rest of us. South West London is a very competitive area though! Totally depends where you live. You are in an easier position if your DD's already at a prep school as their school will be preparing them for 11+ exams and will have loads of advice and suggest suitable schools to sit (most sit a minimum of 3 exams... one you expect to definitely get, one you'd be most happy in and an aspirational school). You need to be happy to go to all of them, otherwise no point sitting them. Lots of kids (especially in SW London) sit more than 3, which is quite sensible really, assuming they are not under pressure. It is not worth over tutoring, but is worth tutoring, especially if your school won't be doing exam practice, you want your DC to end up in a school where they will fit and not drown and be miserable at the bottom all of the time.
I know nothing about grammar schools (as my DC weren't grammar school material), but all the other schools you listed in your original post are schools we have visited/sat exams for, amongst others.
I think this year in particular has been hard as there is so much competition.
You need to visit all the schools, see which ones you and your girls like and go from there. All the schools are different, all look for different things, but there is a private school for everyone.
For the brightest: LEH, PHS, WHS, G&L, KGS
Somewhere in the middle: SWPS, SHS, St. Johns, Epsom, St. George's

I think Sutton High would be worth a look, too. I definitely got the impression that it was seen as less competitive.

Look at schools like More House and Queensgate. They are both very accessible on the tube from SW London and are far more inclusive than some other independent schools.

@cheese10 , they go to the schools I mentioned, a typical average prep dd living in Hinchley Wood (good transport options) will sit SHS, Notre Dame, Tormead and Claremont Fan & probably get 3 offers.

Honestly I'd go for a tutor and have them assessed and see what they recommend. They could be really bright and their school has let them down or that really could be the best they can currently do. My experience of large london primaries is that they often lose the middle. There's a huge emphasis on bringing up the bottom set to pass SATS and proving to ofsted that they are stretching the top. The middle is just left to it as their aren't enough resources.

It's also about ethos and what appeals to you. Do you want them pushed? Do you want a very broad curriculum? There are so many factors really.

Not sure why you’d have an expectation of a state primary having any knowledge of private secondary schools?? Your emoji indicates some irritation.

Include some "back up" schools - Claremont Fan, Radnor House, Ewell Castle, St Cath's Twickenham, Notre Dame Cobham etc. Further out St Teresa Effingham & Tormead. You need a range especially if you are after 2 places. St John's is very popular & difficult to predict entry, if you include it in your final selection you need 3 more schools on your list with 2 that are less competitive.

How old are they and are you willing to separate them?

There are two, long threads covering those schools this year which you could read:
First one
Second one

And if you’re at a prep, then your Head will help. I understand that’s not much use if you’re at a state primary, that’s why I found those threads so useful.