Well, this one has a happy ending. 

The solution was very simple. 

I have a Bluetooth mouse and a Bluetooth keyboard. 

If I turn off the Bluetooth mouse before going to sleep, I can leave the computer on all night and the screen saver will kick in, and then the Energy Saver setting for the monitor takes effect, and in the morning the screen is off. Pretty sure it stayed off all night (evidence -- not being woken up by it). 

There is a setting that tells the Mac to not be activated by Bluetooth devices, but I'd rather not have to attach a non-Bluetooth mouse to the computer to activate the screen in the morning.  

So if you're finding that you can't turn off the screen of the Mac, and have it stay off, try turning off your Bluetooth mouse if you have one.  

And maybe Bluetooth mice should go to sleep and stay asleep if no one is moving them.