Looks like I have this solved. The problem was that the system was locked in manual mode. The button wasn’t showing me the state, but rather indicating that manual mode was “on” and I should push it to turn it “off”.
Further examination showed the root cause. I had operated under the assumption that the system behaved the same as a commercial sprinkler controller – you set your manual times, turn to “run”, and when it completes that cycle, the unit reverts back to “program” mode. So whenever I rebooted the unit, it would come up and run the programs. Then I would do something manually, and it would remain in “manual” mode – thus, not running the programs.
Now that we have this figured out, we know what to avoid. Thanks for the earlier suggestions.