Just came in, somewhat cold but triumphant! It starts again. Many thanks to everyone who helped with their suggestions and feedback.
@stykthyn and
@90sickfox got me on the right track. It turned out to be the clutch safety switch. The switch part was fine, but there's a little plastic clip on one end of the rod that goes through the switch. It's supposed to allow the switch to be adjusted to your clutch pedal's play. In my case, it was "adjusted" all the way out, and moved freely up and down the rod. That meant that the actual switch part couldn't engage no matter how hard you pushed the clutch in. I ended up replacing the part, which I probably could have rigged in some way to get working, but oh well. There's no sense rigging up some half-bad workaround that fails on you at just the wrong moment.
I also happen to have a multimeter that can't read volts. That made for some head-scratching.
Here's a tip for diagnosing this problem that I haven't seen discussed anywhere:
I have one of those logic-safe test lights from Harbor Freight that lights up green when you touch a ground, and red when you touch 12V power. When poking at the connectors for the clutch safety switch, I discovered that the ignition switch connects the starter wire to ground when the key isn't in the start position. This made it easy to check my wiring after replacing the switch. I took the signal wire off the starter solenoid and hooked it to my probe, propped it so I could see it inside the car, and then went inside the car and pushed in the clutch. When the new switch engaged, my test light lit green, indicating a good ground. At that point, I knew I had a good connection going from the starter solenoid all the way back to the ignition switch.
Had I known that at the beginning, I could have identified a wiring issue right away, instead of spending quality time with the starter and starter solenoid, and wondering how in the world my battery could run the headlights fine while measuring 0V.
So, tomorrow I get to put the dash back together (because I needed the practice, doncha know) and drive it over to finish up this defective clutch thing.