Based on all this I think it is your transmission, that it is jamming in agitation and that is what is causing your problem.

As far as repair is concerned, you can purchase this same machine, a basic Whirlpool top loader for about $300 to $350.00 in our area, sometimes even less if it is on sale. The same model is sold under the Roper and Estate brands, and sometimes under the Admiral brand usually at an even lower price.

If you want to continue to play with it, see if there is a used appliance store in your area and they might have a used transmission they have sitting around that you can purchase inexpensively. The transmission comes out easily, three big screws undone and there you are.

Tech from APP suggested that it sounds like the electronic control board, so I ordered one and waiting for it to arrive. Does this sound reasonable, based on the newly discovered fact that the motor seems to run fine during the fast and slow "t4" test, but only hums at medium speed during the test?