You are correct on the working on the rear end. If you don't have a lift I would say forget it. You back will never be the same. Trying to remove the center-section and replacing it - ABOUT 6 TIMES is rough. If you are a young man fine over 40 better get some help. I listening to the video it sounds like the backlash is excessive between the ring gear and pinion. Easy enough to check while you do R&R on the U-joints. Remove the differential cover and leave 1 or 2 bolts in at the top and drain the gear oil. If your cover has a magnet look for excessive metal particles. If you see big pieces of metal it's broken. Small fine almost powder like is OK just not a ton. To get a clean reading on a dial indicator you need to spray brake cleaner inside on the ring and pinion. Don't want lube oil to throw off your readings. You need a dial indicator with a magnetic base. Tons of YouTube videos on how to do this. This is just one I usedThe whole rear end assembly is not in the cards for now, I’ll do that later, once it’s painted. For now I will swap the U joints and try to find someplace to fix the rear, that doesn’t seem very DIY from watching over the years.
Once you get a reading you will know if you have excessive backlash. There are 4 bearings inside. 2 carrier left and right and 2 on the pinion. If you have a LSD this may also be an issue. I have no experience with them except installing a new TrueTrac. Rock the ring gear with you hand and excessive force is not needed to get a reading. I would take 2-3 readings along the ring gear. If excessive backlash you found where your noise is coming from.