You did a fine job cleaning up that ‘third member’.
Prior to installation a couple of things that I would do, just to have confidence that the diff is in good shape:
1) put a torque wrench on the end caps, pinion nut and adjuster stay bolts. Check them to the factory assembly spec for torque, they should not move.
2) use a dial indicator and check lash at the ring gear. Verify it is with in the factory limits,
3) use yellow gear paste ( the exact name escapes me at this senior moment) for contact pattern between the ring and pinion. Verify that the pattern looks as it should considering the age on the gears.
You had said that the gears spin freely, but unless you really know a lot about this particular unit, I would want to get as much information as possible prior to installing it. If it does have a groan or a whine, at least you know some of the key set up dimension.