Finally getting time to reply. Yes mine has the rear seat controls, and no they're not working either.I am not much help. My 04 Rainier and my MIL's 06 both have that 'deluxe' thermostat style system you have.
Mine has eaten a few blend doors and the MIL's melted the plug where the power goes into the blower fan but the displays haven't exhibited the same problems you have.
Do you have rear seat HVAC? Does that work?
View attachment 192131All Wiring Diagrams for Buick Rainier 2006 model – Wiring diagrams for cars
AIR CONDITIONING Automatic A/C Wiring Diagram (1 of 2) for Buick Rainier 2006List of elements for Automatic A/C Wiring Diagram (1 of 2) for Buick Rainier 2006:(i/p harness, 15.5 cm from reci- rculation actuator connector)(i/p harness, 16 cm from sp201 breakout, toward dlc) s211(i/p harness, 5 cm...portal-diagnostov.com
Update though, as of today I'm actually getting heat out of the vents, I just can't control it. Starting to seem more like the blower motor resistor. I tried a junkyard unit prior, but no way to tell if it was working or not. Just a weird issue since most problems with these climate control systems the digital display is lit up.