Depending on how you register the domain, there are some registrars that require no info at all. One of those paid with Monero creates no links to your identity.
But yes, self-hosting does not shield you from court orders. If they find you they can still access your shit, depending on how much your country's infosec police gives a shit and/or how closely they cooperate with US agencies.
Yeah but we're talking about API tools, not web browsers. People are using postman to see JSON, XML, or whatever horrible format the devs on the other side chose to use, not to render HTML graphically. If you query an HTML page using curl, you should get the HTML back, I don't see what's the problem