If so I would expect such sites to appear, and the only way to secure a child device is to have a whitelist of webpages (to avoid the proxies), putting us back close to where we are today.
The problem is that it's easy to make, easy to deploy, easy to make money on, and a single site opens up the whole Internet. It will happen even if it's illegal.
Compare this to adult webpages setting the header. They will probably be quite willing to do so, since they want to make their money legally, and there is probably little money in serving to kids anyway. And even if a single out of thousand adult webpages refuses, it still only opens that single site.
Worrying about the latter makes no sense, because they are sort of like organized crime. People still take drugs even though they are illegal.