Then we probably agree more than we had realised, and I assume that OP, whose post refers to private schools who allocate places through competitive entrance exams, with a fallback of a state school who rations places through the mechanism of religious practice.
Many people will take a place, if offered, at a properly comprehensive, comprehensive school. There are lots of advantages. Local, free, representative of the community our children are growing up in, scope for bright kids to do well, and room for the average and below average. The issue arises when that option is not available. The assumption is often that you will somehow find the money and pay, and many independent schools expect to take around 60% from state primaries. The problem though is that it is not that straightforward.
We hit the barrier at primary stage when the local authority was hit by a sudden surge in demand in our area and struggled to provide places for all their four year olds, and it became clear that this was going to be a problem throughout. Our local secondary was in special measures, and frankly failed to provide an acceptable level of education. (I mentored a bright kids who had a shockingly narrow set of GCSE options.) However we soon discovered that some private schools were trying to sell us things we did not want to buy (poshness etc - though I liked the idea of not having school on a Friday afternoon to allow parents, and possibly teachers, to escape to their country home). Then came the problem of actually getting a place, with all sorts of tests at 3+.
That bulge has fed through to secondary. If you are thinking of a private fall back, particularly if you have boys, live in SW London, and your child is of average ability, it is quite difficult.
Of course there are alternatives: moving; attending church, tutoring, which increase your chances of a place at a sought after state school. But my assumption was that OP was asking about alternative Independent options now that traditional fall backs like Emanuel and Ibstock are increasingly popular and becoming more selective.
South London does not really have the same problem. The Kent Grammars, Wallington, Graveney etc, top slice quite a lot of the brighter children in the cohort, as do the Dulwich schools, meaning that other private schools in the area often have quite a broad intake.