I now kind of feel that I wasn’t very clear in my feature request. I spent a lot of time justifying exactly why I wanted it, and comparing it to other alternatives, and so forth.
The reason why I feel I wasn’t clear in what I was asking for was that this feature request and my feature request are all but identical! (With some slight implementation differences of in what part of the UI you put the URL.) I don’t think there’s any difference that even matters: it’s ultimately asking for the same thing.
Given that Beeminder already has the custom app functionality that @adamwolf mentioned in that thread, the need for this exact feature became less necessary for me because I can get something pretty close that way.
It still would be nice to be able to do the thing that I described/the thing @narthur describes here. Having the ability to enter a simple URL in the Beeminder UI somewhere for it to fetch data from is simpler than setting up a custom app, so for simple use cases (like what @narthur describes above) this would be nice to have.