ThinkingGobling, the % of private school parents doing what you do here is going to be absolutely tiny.
The majority will suck up the increase in fees (which will probably be less than expected, because it does look like private schools across the UK are finally waking up to the need to be more frugal in the way they run schools, after years of wasteful spending on things like souped-up facilities and taking parents for granted).
A minority will drift away from private schooling (mostly at primary level) or not start in the first place, but the figures won't be massive and by the time we start to see private school kids migrating to the state sector, the numbers of pupils in general will be on the decline for demographic reasons and the kids will probably be (mildly) welcomed.
Nothing very exciting will happen either way.
By the way, kids have been shifting from private to state in a small way already, over the past half decade. I see the 7% figure being bandied about here, but it's actually only just over 6% now.
I actually use a private secondary school in another (rich) country (following a government-run elementary school), and it's affordable. I find the fees at UK private schools - and the rises in fees - pretty shocking and feel that there has been astonishing complacency about this for decades. A bit of belt-tightening and fat-trimming might not go amiss.
Mostly likely we will see private schools adopt different strategies - some will shift towards being more affordable schools aimed mainly at parents who have missed a good state school place, some will focus on being nurturing places for kids with SEN, and some will double down and become luxurious money pits for the truly rich.