Well, a few updates before bed.
Since the entire assembly came intact, I originally decided it would be a simpler task because there would be no need to remove the brake assembly and could instead set the entire thing down into the base, reconnect the springs, then align and lower the outer tub onto the assembly and be good to go. But, that's not how things worked out.
First, I couldn't get the basket hub block off to save my life, even with locking a pipe wrench and pinging it with a hammer. Then, since I'm not intending for this to be a super strict "by the books" repair because I'm not needing it to be a daily driver, I figured if I could at least turn the basket hub enough that it lines with the conveniently shaped grooves in the outer tub opening, I could lower the tub down into place and as long as the machine works fine, it doesn't matter if the hub is removed and replaced. Normally, up until this point, I could always grab the basket and turn it by hand and even with the brake engaged it would "slide" a bit. So having no luck turning just the hub and shaft on their own, I temporarily slid the basket down, bolted it hand tight, and tried to turn clockwise. Not a budge. Instead of the shaft turning as it should, the entire trans assembly moved so much that I had to readjust one of the springs at it had almost popped loose. It's like the brake and transmission are cemented in place.
So then I thought, maybe if I reassemble the thrust bearings and pulley, slip the belt on, and turn the motor on for spin just briefly enough to disengage the brake, I can then turn the hub more easily to line it up. Easy right? No. The basket hub and shaft won't spin at all. At all. Turning the pulley by hand is stiff and the brake won't budge. But it was a relief that I used a medicine syringe to put enough pressure in the water level tube for the machine to start agitating, and it agitates perfectly, and doesn't even have that "wahwahwah" sound the other one did while dry agitating. So I don't think this "new" transmission is to blame.
Just for kicks, I took the transmission pulley and thrust bearing assembly back off, and put it in the same order onto the old trans assembly sitting over against the wall, and viola! Turning counterclockwise by hand gracefully releases the brake and the transmission begins to turn, while clockwise starts to oscillate the shaft (because as I may have said before, the old transmission seems perfectly fine from an operational standpoint, but with slinging oil for 3-4+ years there's no telling what damage to the gear system was done).
So while I would adore any advice and anyone who has an explanation for this predicament, my theory so far is this:
-the original brake assembly from the old trans is in perfect shape, probably because everything soaked in the gearcase oil has been "preserved" in a sense.
-since I'm using the thrust bearing assembly from a different machine, it isn't "calibrated" to this new brake assembly, thus, it may have needed a second washer on the pulley for the cam to be able to push the brake free on its original machine that I just don't have here.
-while this assembly came completely intact, even with gaskets in great condition, the seller may have just kept everything together despite a problem with the brake because the frame acted as a shield for storing/transporting the transmission itself. I only paid for a good transmission, so maybe they figured anyone buying it would know how to either adjust the existing brake or would just trash it anyway because of already having a functional one.
So despite thinking I might not have to tear down the brake at all, tomorrow's project is to remove my original brake rotor and stator, clean them, use the new snubber ring I bought today because why not, and transplant it to the new transmission, since I know the thrust bearing assembly I have now is perfectly fine. I even bought the self-drilling 1-1/2 inch screws for removing the brake and spring today, so I should be all set. After that, hopefully getting the tub, basket, and agitator back together will be cake, and I'll be able to do a few trial runs before the night is over.