I'm afraid that you want to simplify the issue beyond all reality to make a false argument.

I'm afraid you want to keep your head in the sand about the real reasons that rich and powerful people send their children to private schools Andy, you'll dig up any objection you can to the argument that people with money choose private education to acquire an enhanced future for their children.  What does it matter if 1/4 of those attending the private school that you attended were there because their parents were abroad? 3/4 of them weren't! And would those who were working abroad have chosen state schools over the private schools had they been based in Britain?  What if state schools had been available with a number of boarding places, would they have chosen those schools?  There is no reason why the state can't provide a system where boarding costs are paid for by the parent but schooling is provided in the normal state system.

So, yes, at the moment, there will be some situations in which children will have to go to private boarding schools, but no, this is not the major reason that people send their children to private schools.  To use it as such, is a ****-poor attempt to obscure the truth rather than an attempt to elucidate it.