Well, I guess I've figured out the problem. Duh.
My monitoring configuration has some limitations. That's because I'm using the computer that's uploading the stream to a streaming server as the stream playback platform and a second computer (on the same LAN) to run the Pira monitor app (with an audio cable between them). So -- when the local Internet goes down or has some kind of hiccup, yes, the stream stops and, yes, the alarm 'sounds', but an alert email can't get out -- because the Internet is down.

I guess the way to do this properly is to have an off-site computer(s) handle the stream monitoring. Lesson learned.