Oh, this is beautiful.
I realized that the machine in question wasn't connected to the VPN, so it could not talk to the on-premise console. So I started the VPN on that machine.
I also realized I had five logged-in instances of the console open--apparently, though some of them may have timed out. I closed them all and logged into the console again, where I received two new alerts from the affected machine, a couple minutes apart.
I also received two email alerts of these new detections. I ran the email test in the console after each one, and it STILL failed, both times.
So I have to conclude that the email test is buggy.