Yes, exactly, this should be really simple as a foundation: by default the Internet is for adults. All of it. Where it's desirable for things to be available for kids, create the economic incentive and easy tools for parents to use and run on a white list, not a black list.
I'd actually go somewhat further though and ask whether it's a good idea to even do this via web pages at all. We have a great potential system for this already: DNS. Do something useful amongst all the ridiculous vanity and spam TLDs for once and set up a ".kids" gTLD, or ccTLD for that matter so that different countries can set their own regulatory standards naturally (ie, .kids.us, .kids.uk etc). Domains could also be used for some broad buckets for people who don't want to drill in, ie, .1-6.kids, .7-12.kids, .13-17.kids, or whatever is deemed appropriate, but simple age brackets that would offer some sane defaults. 1-6 could simply not allow any ads, user generated content or algorithmic feeds whatsoever for example. There are a lot of knobs to turn. And then at the registry level it can be ensured from the get-go that anyone getting a .kids domain is fully identified, located in the country in question, has valid ID, has specific credentials or is an accredited organization, or whatever other criteria makes sense.
But ultimately the point would be to create something that is built right from the ground up, and in turn that doesn't interfere with what has already been built at all. Something that can also be worked with at the gateway and thus cover every device on a LAN, and for that matter can easily be plugged into the vast number of powerful tools we have for working with that stuff. It'd be easy to put a nice UI on all this, even to make it higly automated. For example, have a setup wizard where you enter children, put in date of birth for each, and it'll spit out a password for each one. This then auto-provisions the network such that each kid has their own VLAN (password for PPSK or even wired connection) and is automatically limited to the domain groups of their age bracket, which then changes as their age changes.
Parents should be able to dig further in and get more granular with content categories, metadata for which could be required for anyone hosting a site within that domain, but I think there is the potential to make something both pretty bullet proof and pretty accessible, using existing tech stacks, and without impinging on the present internet at all including privacy and anonymity.