The rest were attempts to stop the audio interference I was hearing. Tried new audio drivers, new video drivers, new chipset drivers, et cetera, et cetera, et cetera. No matter what I did, it kept producing white noise anytime anything happened on the screen. I did some more research, and finally hit upon the correct Google search phrase. It turns out my on-board audio isn't actually a set of chips as I'd thought. It's a software emulation of a set of chips, and the emulator runs as a process on the CPU, which explains why it's so easy for interference to happen.
Turned off the machine, opened up the case, and installed the old Sound Blaster Live card that I'd had in Speedy. Some rethreading of sound cables needed doing, along with moving the rear fan to the next slot over. On the plus side, the card has a connector for the digital sound cable from the burner, which the on-board audio didn't, so I ran that cable, too. Closed up the case, fired the machine up, and reinstalled one last time. Ding, dong, the noise is gone. Woo hoo!
Now to start putting in some more apps.