My replacement display board arrived today. It was a snap to install and the Vario is working properly now.
I disassembled the old display unit and found an obvious cold solder joint on one of the display leads:

The cold solder joint is on the left-most lead of the display. You rarely see such an obvious example of a cold solder joint. Usually they're more subtle.
I would speculate that the lead made adequate contact with the solder when it left the factory and for the first couple of months of use. After a lot of vibration from grinding, the contact area probably became intermittent, so that when the grinder was running the lead would momentarily lose contact with the solder. I would further speculate that momentary loss of contact on this lead causes the display to blank, and it will stay that way until initialized by the CPU, which evidently is done only at startup after plugging in the grinder.
Here's the rest of the board:

Note the coffee grounds between the leads of some switches and the white oxidation on some solder joints. These are also candidates for cause of the problem, but less likely in my opinion.
If my replacement display ever fails, then I'll touch my soldering iron to the cold solder joint, which will fix it, and I'll wipe down the board with isopropyl alcohol. I'm pretty sure the display will work fine after that.