Your video actually has the opposite effect: it shows that there are no (easy) ways for parents to protect the children and government regulation is necessary.
The UI should be a single checkbox named "child mode" and small, gray link "configure" that opens the menu from the video. The parent just clicks the checkbox, and sets up a password, a fingerprint or paired device to unlock the mode. That's all. After this only government-approved apps and websites can be opened (parent may edit the white list if they are ok with navigating through a maze of ten thousand menus) and children are perfectly safe. Government may white-list only apps and sites that implement measures for content safety and may not add non-compliant apps.
Also I hate watching vertical videos on a laptop.