My wife works for a major private school and knows many teachers, heads, etc in the field and can honestly say state school applicants are 100% welcome. She knows staff at all 3 schools you mentioned in your post. On no level is there a preference for private pre-prep school intake over state at 7+. (Or prep over primary at 11+ or private senior over state secondary at 16+)!!
One must still pass the entrance exams, and perform well in interview of course.
OP, a bright, confident Y2 state primary child who has been prepped for the 7+ either through private tutor or by parents using Bond books, past exam papers, etc , should have a good shot at any selective private school.
I back what others have said here, like @2minutesplease
It’s also pointless comparing state and private, offensive even, the difference in spending per pupil makes any comparison meaningless, like sport provision, class sizes, etc.
Many parents have gone the state till 7+ route with great success, as others have said, you must do the prep yourself, but even the best private pre-prep schools in London won’t do everything for you anyway. Keep in mind many parents find this a selfish route as it blocks 4+ places at the best state primary schools for wealthy parents who up and leave in Y3.