Guys, thanks for the replies.
I decided to pull the engine out of the car, split the block from the pan and pull the transmission in order to take a look at it.
Through a process of the above, and speaking to Dave at Chaffins (he's very helpful, and generous with his time), I was able to work out that the brake drum is missing one of its two bushing - there's only one installed, at the end closest to the flywheel, and it's kind of galled. Dave said some builders just choose to do it that way because the drive plate bushing carries the other end of the drum assembly on the transmission shaft. Except when it doesn't, as is the case with my unit.
The transmission shaft is a bit ripped up.
I dropped the center crankshaft journal, just to take a look. I seem to remember the previous owner saying that the engine had been recently overhauled, but he was at the age where 'recent' could have been 20 years ago. As it stood, the babbitt looked decent, and looking up inside the engine, there are still cross-hatchings visible in the cylinders. Based on the power this engine makes, and the components inside (aluminum pistons and oil dippers), I don't think it was long ago someone was in here. I wonder if someone rebuilt the engine and left the transmission well alone on the basis that it worked OK at that point in time.
The crankcase oil tube was absolutely filled with crud.
I pulled it, blew it out and reinstalled.
My magneto pickup external oil feed line is also plugged solid. I've ordered a Texas T oiler to install whilst the hogshead is out.
So my plan is this - fit new brake drum bushings (two, this time), polish up the transmission shaft on a lathe, reinstall the lot and see how we go. Gonna miss the Christmas lights, but hey.
I will inspect the drums and brake drum shaft for cracks, thanks Adam. The brake drum is a newer replacement (has the clutch shoes found on the late-model cars).