So Comcast showed just before 10 am. The guy walked into my house, took one look at the old 3Com Sharkfin modem and said, "Let me get you a real modem." He replaced it, enabled it, and now it's running. I'm going to keep an eye on it for random disconnects, but if the modem was indeed the problem, it's fixed now.
He also checked the line into the house, just to be complete. And he had somebody who works at the office along with him, too. I'm not sure if she was just on a ride-along, or there because I've been a problem customer in the past. For whatever reason, I'm pleased at the moment. I hope they don't charge me for the visit, but I'm not holding my breath.