“I should be able to call the API from a web browser just by typing stuff in.”

Ah, you can go much further though – just make it so the same endpoints that you use for your website are the endpoints you use for your API.

One of these days, just like the Clojureists have done with Lisp, people will realise that the Semantic Web people knew what the fuck they were talking about all along. I fear that path is long and dull though.