I was able to test the power button all the way to the inside of the ribbon cable connector. The power button goes to the first and fourth pins on one side of the connector. With the ohm meter on the ribbon cable connector, pushing the power button closes those pins. That tells me that the power button is good all the way through the connector.
This next part is a bit of a guess. At the solder pads on the logic board, I found continuity from one pad to one of the pins on the logic board socket (the first or fourth pin on one side). I could not get continuity from the other solder pad to a pin on the socket. This might mean there is some failure on the logic board.
I'm going to run the computer without shutting it down. As long as the system is always running or goes to sleep, I might be ok for quite a while. However, some day I'll need to hit the power button again. I might try to solder the power button directly to the solder pads on the logic board, obviously a risky solution. I've replaced capacitors on motherboards before, so maybe this is possible.
Maybe I'll look for a top case just in case some part of my testing was flawed.
Thanks for the tip about bridging the pads on the logic board! How awesome!