What test can I do to my washer's gearbox to determine whether or not it is bad?
I decided to go ahead and buy a replacement gearbox.  Testing the new gearbox before I installed it, I determined that the answer to my question is yes, my test does mean that the gearbox is bad.  With the new gearbox, I was not able to spin the input shaft and hold the output shaft still at the same time.

I installed the new gearbox, and now the washer works again.