What makes you think it isn't privately owned? It's true that the domain can't be bought, but you still hit a Verizon server when you go there.

And if you're using an iDevice, you might as well go to http://captive.apple.com, which was designed for the purpose of authenticating to hotspots and Apple already knows who you are anyway. :)

https://example.com works, so it won't serve all same purposes.