Here’s how to check the transmission by hand. I forget which way to turn for agitate and spin but you should be easily able to figure that out.
Try to turn the main inner spin basket by hand, it should turn nice and smooth and easy in one direction and total lock if you try to turn in the opposite direction. Now if that test is successfull meaning the clutch/bearing or clutch/spring hasn’t failed then the direction that the tub turns is the direction it should be turning when spinning.
Now that you know what direction to turn it for spin, try to turn the transmission pulley by hand in that direction, the pulley should be locked and turn the whole complete transmission and spin basket with the pulley.
Now try to turn the pulley in the opposite direction, (which should be agitate), the pulley should turn real easy and the transmission should stay stationary and the agitator should be going back and forth, (very slowly of coarse since you can’t turn it fast by hand — you might need someone else to watch and make sure the agitator is moving). If the pulley locks up or is very hard to turn by hand in the agitate direction then you have a bad transmission.
If this test seems to work ok then you will most likely be ok with replacing the idler pulley and belt. This should give you a couple more years use out of the machine if the main tub seal or the lower clutching bearing or spring doesn’t fail.