I don’t think that is true.
Having worked in education in a place with excellent state schools (even the ‘worst’ would be sought after in much of the country) but also multiple private schools, I don’t think it’s the case that the presence of a good state schools means nobody supports private.
There are some families that simply don’t consider state - at an extreme, the Royal family will not use a state school, and there are other families that simply would not consider it. Those who have been privately educated themselves, and have peer whose children all attend private, automatically choose private schools as ‘the option for people like us’.
Even within an ecosystem of extremely good state schools, there is a heirarchy. Where ‘the current best’ school
is unavailable to them, some parents will choose a private school instead of the ‘genuinely very good but not quite as good as X’ school.
Equally, some parents just want a school of their active choice, not one allocated by the admissions process - they may want a small school, or single sex, or that plays rugby, or that has very small classes, or that doesn’t have their child’s feared peer from nursery, or that offers flexi boarding, or longer days, and want to pay in order to guarantee they get their choice.
Some parents find the socially non-selective nature of comprehensive schooling quite difficult, and wish to protect their child by buying an education where they will be in a slightly separate bubble.
Some parents are looking for a school to provide eg high level music, sport, drama etc within the school day (rather than eg finding county or club provision out of school, which can be hard for logistical reasons). Some private schools offer these.
Some parents are looking to gain an advantage in the planned next stage of education- in the same way as some parents move their children to state for 6th form in the hope of advantage in university admissions, there is, in selective areas, an industry of ‘grammar crammer’ primaries offering the prep for 11+ that state schools are not allowed to offer.
None of the above is an argument for VAT-free private schooling, but it does explain why private schools flourish and are chosen by parents even where state schools are excellent.