The agitator will not turn manually (with the motor/coupler disconnected.) Does this mean it needs a new gearbox?
No it does mean that it needs a new gearbox.
If I wanted to test the gearbox I would remove it from the washer first.
Then I would turn it by hand the same place the motor coupler would turn it on the short side shaft.
I would turn it one direction to see if the shaft agitates back and forth.
Then I would turn it the other direction to see if the clutch spins.
But before I did any of that if it had a broken couple I would slap a new motor coupler on there and give it a try. 99% of the time when I find the motor coupler broken all I need to do is replace the motor coupler.