I could see what he says happening here.
Here we have good comps and a few outlying super-selectives.
Even though the comps are good , mc parents like to tutor and have a go at the super-selectives, for whatever reason. (competitive atmosphere, kudos, high performing peers) but as I say the comps are good and send kids to good unis etc. So when they don't make it to the super-selectives they are by and large happy with the comps. There is no problem with telling people that your child is on the comp, and no-one talks about what sets they are in.
However, if the top sets in the comps are skimmed off into new grammars, I very much doubt that these parents will be happy to send their children to what will be downgraded comps: everyone will know that they failed the 11+, they will worry about the cohort in the comp without the top sets, and my guess is that the mc parents of just-missed-grammar and middle ability kids will indeed head Private if they possibly can. In London if they have been in their houses for 10 years or so they will release the equity
In my borough .the proportion of parents using private schools has shrunk - not because of the struggling finances of the mc (it's gentrification central) but because the local comps are good and have a good social mix. High FSM, a critical mass of bright kids (from all socio-economic groups) a critical mass of mc kids (of all ability ranges). If the bright kids are taken out in greater numbers than the tiny few who go to super-selective, the MC will flee.
This is what Neil Roskilly is saying. I think he is right.
I know London is different , but this effect could happen in many places. Perhaps more so because people outside London do not have such high living costs, and neither do schools, so more leeway to decide to find money for school fees.