Quick update: Today I had to reboot the machine as per instructions of a malware removal tool. The last couple of weeks it's been repetitively hit by a couple "update your drivers" attacks, and the big 3 - SuperAntiSpyware, MalwareBytes, and AdwCleaner - aren't taking it out.
Once again I couldn't get it to boot. It would boot a DVD but failed consistently on disk, and of course I couldn't see what was going on.
But the video card arrived this morning. The only problem is that its HDMI connection is different than my cable's plug. I don't know the terminology, but the card has one side square, whereas my cable has both sides flaring outward into a trapezoid. So I'm down to one monitor. But it instantly booted - no beeps, no problems, completely stable.
I was gone all day, and still have to catch up on a bunch of stuff before I can experiment with the setup. Clearing CMOS on this Intel board requires removing the battery for some number of minutes - there's surprisingly no jumper.
We'll give it a try later or perhaps tomorrow. But right now the machine is perfectly fine with the constraint of one monitor.