White-listed sites could report the content rating (for example in HTTP headers) and the browser in "child mode" would refuse to display non-appropriate content. Lack of the header means that content is for adults only (unless parents manually white list the site). Also, websites could use different domain for "child mode" version.

> Would this not immediately lead to demands for hardware makers to prevent the installation of non-compliant operating systems?

In child mode, yes, installation of any other OS (compliant or not) should be prevented.