I am aware of a number of efforts to replace not just the web, but most of the services on the internet, with private intranets to serve specific subgroups. The goal is to increase security, privacy, and to regain the useful things that the internet is increasingly unable to support well.
I suspect that sort of thing will become increasing common.
I can't, because none of them have a public point of presence on the internet. That's rather part of the point of doing it -- to keep the internet at arm's length. They're all, effectively, invite-only since they serve groups of people who are already known to each other.
Some do this on a small scale, like operating a VPN and a series of services accessible over the VPN. Others on a larger scale, like running what amounts to a small ISP and communal servers.
If you're into the idea, it isn't be that hard to get things going on the technical side, if you use the internet itself as your "dumb pipe communications channel". You just need some friends who are interested in joining you.
I am doing something like this myself. SSH as the access method. Invite only. Quality not quantity. Member get member, so to speak. I can’t sit back and complain and then not do anything.
Not to these subgroups, no. They don't provide the benefits of security/privacy, are walled gardens, are considered by many to be problems, not solutions, and are services on the public internet. The whole point of these efforts is to avoid being on the public internet.
I suspect that sort of thing will become increasing common.