I took apart the right-hand switch, opened up the headlight and checked some connections. All the connections tested clean (0 mV, so, connected). I shorted the starter relay to send power to the starter motor and the bike started up no problem.
After I did that, the starter button started working again. So I probably have a loose connection somewhere.
I threw a nice fat wire in my under-seat-toolkit. If the problem comes back I can just bypass the relay again.
One problem: when I opened up the starter-button/throttle/kill-switch housing, there was a metal bracket holding the starter switch in place. I didn't take a picture of how it was assembled and I couldn't figure it out when I went to put it back together, so I just left it out.
I looked for some microfiche to tell me how this goes together, but no luck. I think it's probably there to reinforce the plastic of the switch against gradual vibration damage, or something. I think it should be fine for a while, but if someone can tell me how it's supposed to go in I would appreciate it!
Here's the part, and the starter switch side of the assembly where it's supposed to go.