Since your Avalanche had the green connector it could have been prewired? OR just your column wiring harness was the one for a heated steering wheel and dash harness did not support the heated wheel? Too late now, but check and see if you have a fuse installed in the under hood fuse block, #48, 10A HTD STR/WHL. This may help some one else that finds they have the green connector in the vehicle.
The 12v wire for the green connector goes first to the heated steering wheel module (in the steering wheel). You have to select a 12v source that is "hot" only when the key is in "run" or "acc" or your battery will go dead in a few hours even if the switch for the heated wheel is off. Yes, you covered this too.
Steering wheel nut 21mm or 13/16" should fit.
I prefer to use a steering wheel puller. I worry the aluminum steering wheel shaft hole would egg shape, rocking the wheel to remove it.
For the tilt lever removal, I just pulled it off, yanking by hand to the left.
There was an airbag change in 2013, hence a SIR coil change too. Connectors are different for the airbag. Connector colors changed too. Part number you listed is for 2007-2012. New part number for 2013s 22911594
I totally agree with your replacing worn parts while you have it apart, and doing a professional wiring job.
The only other thing I can add; if disconnecting the battery (I did so the horn would not "honk" while I tried to get the airbag released) is move your seat all the way back if it is power First.