I did some more research today, and compared my script to another client's checkout, whom I also wrote some ajax script for, which phones home to their server. What I found was that the previous script that was once working, is now no longer working either. I know this script was working at one time, but now it's behaving like the script I'm writing now, i.e. simply doesn't seem to be firing.
I want to emphasize this point: the ajax script is producing zero errors in the console, nor is it showing any requests being made to a URL. It is as though the script is not there at all.
Is there anything in the checkout which might be preventing custom ajax calls from firing?