Qbean There is a difference in selectivity amongst the private schools you list. If you were in a Prep School they would advise you to try for one aspirational, a couple that you should get into and a less selective one as insurance, just as insurance against bad days and to be sure of a choice, and often that choice gets firmed as a result of the experience of the entrance exams and interviews. They would advise you about the match between that and your DD's ability. I know that is hard when you are coming from a state school (been there, done that) but fair to say they would have to be superbright to get SPGS, lots of superbright supercharged trying for it, then LEH which gets the superbright from Surrey who wouldn't travel to SPGS, but also lots of down to earth local state school girls (and not all supertutored as the parental Chinese whispers might suggest, as a guide they say they usually find that their girls score in the top 5% in reasoning tests - that would be after a handful of practise papers not a year practising at some tutors coalface when it is no longer a reliable indicator of ability, they use untutorable tests), then Latymer Upper which is getting more selective all the time and G&L, which is apparently suffering the competition, then KGS (another getting more selective all the time) and the Highs with perhaps Surbiton being a bit less selective. Not sure where Radnor House is now but St Catherine's a fall back from them all.
However don't get selectivity mixed up with the best school for your DD. All these schools will enable bright girls to get good grades, in arts and sciences, and to the best universities, probably the same outcome for the individual, except that confidence is of course a factor and sometimes being bright in a less selective school will build that better than being surrounded by very bright girls, or sometimes it is built on being with the brightest, depends on personality.
So no there isn't a bunch of bright girls who get all the places, most girls get a range of places at the schools that are right for their level of ability. Then there is a big game of musical chairs when everyone makes their choices, not always choosing the most selective. Waiting lists move depending on which schools find themselves most and least popular in a particular year. You might find this interesting, three years offers and places taken for non selective Old Vic www.oldvicarage-richmond.co.uk/admissions/destination-of-leavers/
The details of the pupil bulge are here, 200 more in 2015 than this year. www.richmondinclusiveschools.org.uk/files/index?folder_id=10529714
And you might find this interesting www.theguardian.com/education/2003/oct/08/schools.uk5
Re getting a chance to join in, all the schools will say they try to be inclusive but bear in mind that some of these schools are winning national competitions in sport, music, whatever so if your child isn't very good or very confidant they may get discouraged. But there are always house competitions etc.