I don't agree with this argument. When parents who value education send their children to state schools, it's good for the school; it helps create an ethos of academic excellence there. If pupils come from families who believe that regular attendance and dedication to study are important, that makes the job of teachers enormously easier. When pupils are surrounded by other pupils who value learning, they learn better. Cultural capital is real, and when parents with cultural capital (not just economic capital) send their children to a state school, the whole school and community benefit. But when these parents decide that the state school isn't good enough for their children, it can become a self-fulfilling prophecy.
Also, schools receive funding for each pupil, so if more students attend the school, the school receives more government funding. That's why state schools are so keen to fill all their places. For every empty place, the school is losing money.
It's not at all like the NHS vs private health care. When a pupil chooses an independent over a state school, that doesn't mean that the state school has more resources to allocate to the remaining pupils; it means it has fewer resources.
It's a sad truth though that state schools are woefully underfunded after so many years of Tory government. Pupils with SEND have suffered in particular. Getting an EHCP is ridiculously difficult as @senmama007 has said, and many pupils with SEND fall between the cracks.
So even though I'm not opposed to the idea of adding VAT to independent school fees (apart from in the case of pupils with SEND!), the real change that needs to happen is for the government to pour more funding into state schools, especially for pupils with SEND. Over my years as a school governor, I have seen financial support for SEND decline more and more. There used to be support available for pupils with varying levels of SEND; now it's an EHCP or nothing (and EHCPs are rare as hen's teeth!). Whether or not a pupil gets an EHCP also depends on how good your school SENDCO is at writing the application 😢