"many local internets"
yeah same here, fixed IP with a domain and a web site, DNS, and other things. Helps with customer projects, too. I can mirror a customer web site on my local LAN server, as an example, fix stuff locally, and say "hey take a look at yourdomain.mydomain.whatever" to see if they like it.
But as for ALSO being "part of the big intarwebs" - why could THAT be a problem? Or is it just the DNS system?
If you hate all that in the current DNS system, just use ".onion". Or create your OWN DNS system [which you'd have to pay for, probably, but it's usually cheap]. As for me I could create as many names as I want if you don't mind "mydonain.whatever" tacked onto the end of every name...
but yeah the existing system is there, relatively unfiltered, reasonably secure, and apparently NO POLITICS INVOLVED [or that's the goal]
/me imagines "Cancel Culture" applied to web DNS registration... *shudder* [and it's my general opinion that a FOR PROFIT corporation is LESS LIKELY to engage in this kind of DISCRIMINATION]