Just wanted to chime in, 2012 Tahoe LTZ here and I'm having the same exact issue as of a couple months ago. Heated wheel works ..... until you start the engine.
I went as far as taking the wheel off, disassembling the entire wheel to get to the control module in the back left side (behind the plastic cover) and opened up the module. Everything looked good, no bad solder joints or obvious signs of damage/blown components. I am at a loss!
I just bought a used Escalade heated steering wheel for practically nothing. I'm gonna tinker with this thing and see if the Escalade wheel works.
I tried to just purchase a whole new steering wheel about a month ago (my P/N is 20998065) and nobody - NOBODY has this wheel available anymore. I called at least 30 different big name dealerships around the US and even a few large OEM parts houses and came up with nothing. Even the guys I spoke with were baffled that nobody, anywhere had these wheels anymore. Plenty of them in other colors though!
Anyway, I'll see what I can figure out and post updates.