The newest models already know whether you’re a bot or not before the checkbox loads. A massive majority of the internet goes through Cloudflare so by the time you land on a site you already have what Cloudflare dubs a Bot Score based on your behavior across the web.
Checking the box really just confirms what they already know. There’s a second form which I’m sure is even more prevalent than the checkbox that renders nothing, requires no user action, but can prevent form submission if you fail the check.
Privatization helps people who can afford to not be in a situation where they are choosing between only shitty options. The point of a (good) public system is that it’s a consistent, good baseline for ALL people regardless of their socio-economic status.
So in your example, sure, if you can afford it you can put your kid into private school or simply move but what about your neighbors? Can they as well? Society functions best when everyone has fair equity.