Then it hits me: the alarm hasn't been working right at all lately, and it controls the starter power from the ignition. So I pull the alarm out, get some electrical tape and hotwired the starter leads together. Try it once and it doesn't turn, but sounded very different. Try #2 yields a slight hesitation and BANG, we have liftoff. I drove it home, but it's too late to be messing with it tonight.
What's the concensus? Did I find the correct problem, or did the real broken part (engine relay, ect.) just happen to start working at that moment?