So I got in touch with the guy who sold me the unit, and he offered a hassle-free swap with a new unit. The new unit was sealed, presumably from the factory. I got home and got the same result straight out of the box.
I've been suspecting that this issue is related to temperature, because it eventually works, and once it does, it continues to work. This time, I removed the battery and held the battery in one hand and the electronics housing part of the unit in the other. Electronics don't care about cold, but batteries do, and presumably whatever sensor is used to monitor heat temperatures does as well. Anyway, after applying hand heat for 30 seconds the unit powered up just fine.
So I'm not dealing with extreme cold temperatures here. I'm talking between 50-65 degrees. Most of the time the unit is in the unheated garage, but I live in a temperate climate. The batteries are charged in-house, and it's probably 65-degrees minimum indoors. So either way we're not talking crazy cold temps. Anybody else have this problem?