Well folks, my ordeal came to an end today. My 14 Sierra 1500 SLT with 1750 miles on it does not shake any more while going down the road. My story:
By the time the truck had about 1000 miles on it, I realized that that the driveline vibration was (1)unacceptable and (2) not going to go away by itself. Took it to the dealer. The tech said that two of the wheels were out of balance. One by 3/4 oz and one by 1.5 oz's. Knowing this to be BS, I went ahead and took the truck. Of course nothing about the ride was different. Brought it back a couple of days later and told the service writer that "this is repair attempt number two - please troubleshoot and repair my problem." Service writer explained that at this point GM has a LOT of driveline vibration issues with this series of trucks and they do not yet have a solution.
He put me in a loaner, a 2015 Yukon XL. I told him "take your time". A week later, he told me to come and get the truck. When I got there, he told me that they had been able to improve the road force variation on the two rear tires by two pounds. BS again. It doesn't take a week to match mount two tires. Not good. I arrived, ready for a spirited discussion and seeing a buyback in my future. At that time, service writer told me that GM tech had just called and asked them to use the electronic vibration analyzer to troubleshoot the problem. I jumped in that sweet loaner and left. Two days later, service writer asked me to return the loaner and drive my truck while waiting for a rear axle assembly to come in. Too much or too little pinion to ring gear backlash will cause problems, but not vibration. I was told that there is a VARIATION in backlash during a ring gear revolution. Wow! That makes sense. The backlash should be equal (or very nearly so) between the pinion and ALL of the ring gear teeth.
Two days ago, I dropped it off to have the rear axle assy replaced. It came in COMPLETE, including rotors and calipers - ready to bolt up. Got my truck back this morning. It now rides like a new truck! During the last visit, all 4 wheels were scratched due to improper tooling being used on the wheel balance machine. I now have four NEW wheels. No pushback whatsoever about the wheels.
Some lesson learned:
The dealers (ALL of them - all brands) do not have the competent techs that they wish they had.
Based on an experience that I had with a 14 Silverado (same size wheel/tires), I was certain that the tires were the issue. Had the dealer done as I insisted in the beginning, the tires would not have helped and they would have been out a lot of money. As frustrating as the first two visits were, I believe that my remaining calm and respectful contributed to these folks continuing in earnest to try to resolve the problem. I'm not happy that they (twice) tried to give me a "let's hope he'll go away and not come back" non solution. I respectfully let this be known. See 1. above. I feel like we now have a good relationship.
I'm grateful to these folks for properly repairing my truck. I really like my truck now.