This Zenith DVD2381 has worked for years, except for a spell a few
years ago when the tray would shudder but not quite open and close.
I managed to free it up.

So tonight, when there was nothing on TV, my wife told me that when
she had tried to play a DVD on Tuesday, it didn't work. She tried
to turn it off, but the light in the power button stayed on. Nothing
appears on the display: no "Hello," no "No disc," no nothing.

We tried unplugging it and plugging it in again, but that did not
help.

The thing still looks pristine inside, so I hate to junk it without
giving repair a shot.

Does this sound like a simple problem to fix?

One not-uncommon failure mode in such devices, is bad electrolytic
capacitors. They may go "high ESR" and fail to filter the power
supply voltages properly, or they may swell and leak (failing to
filter, and possibly damaging the PC board traces).

A decent capacitor ESR tester (which is different than a capacitance
meter) can often identify such caps.

"Re-capping" the device requires identifying the failed caps (or just
identifying "all the usual suspects"), removing, and replacing. If PC
board traces have been damaged by electrolyte leakage, it would be
necessary to rebuild them or bridge over the damaged areas.

It's definitely possible to do. It often isn't done commercially for
lower-value products due to the cost of labor. When you can buy a DVD
player for under $100, spending an hour of technician labor at $100
isn't a great bargain.