I went back to the altar after p/up the note, talking to Delphine, following her up the hill and killing the recently raised dragon by Alduin, and progressing to when I am to meet Delphine again at her place in Riverwood, but I was not able to get the altar to do anything. Do I have to give the horn to the Elders, first? I guess I'll find out after I go back, but if it doesn't work, what's up with that?

I didn't know the alter had a fuction other than to hold the note :/

The Horn of Jurgen Windcaller completes after the Greybeards receive the horn. A Blade in the Dark (The quest where you meet Delphine) technically starts before the previous main quest concludes (same with Diplomatic Immunity).

So yes, you have to return to High Hrothgar and give the horn to the Greybeards.

Thanks, REL_Dovahkin, you are correct. Once I gave up the horn and returned, I received the dragon soul. Too bad that the only way to know that is to have read the Wiki subject, and it wasn't clear even there that the horn had to be returned first.

Thinking on it, since you receive a shout, or an upgrade of one, in my case, it makes sense that there would be a method to obtain a soul to enact said shout. I certainly am not sufficiently logically-equipped to have thought of that myself.

Yeah it's really bizarre that you are expected to figure that one out.