Since most transmissions today are mostly electronically controlled I suspect the drive solenoid might be fried. When you shift into Drive, you are asking the transmission to operate autonomously. In other words, it does all of the work so that you don’t have to shift from gear to gear. In order to set this up, the solenoids that control the way gearing lines up must fire in a certain manner. Most vehicle have two solenoids that set up the gearing. It might be that solenoid 1 is used for first gear; solenoid 2 for second gear; solenoid 1-2 for third gear; solenoid 2-1 (not the same electronically) for fourth gear; no solenoids for Drive, and a special bypass for Reverse.
So, what I suspect is happening is this: when the Drive setting is selected, instead of activating both solenoids to get out of the way — if that is the way the electronics set up in your transmission — the switch is failing to energize them and your SUV isn’t shifting into Drive. To fix the problem, I would recommend having both of the solenoids replaced — one is external to the valve body and the other is internal — as a starting point. This will cost you about $500 or so which is relatively inexpensive for anything transmisson-related.
Now it is possible that the Drive activator switch itself is gone, which is even better because replacing the switch, itself, is about a $300 fix, depending on the amount of labor needed to get into the console to access the switch. The switch is located within the console and is positioned so that when you shift into Drive, the gearshift activates the switch and the solenoids fire. Have your technician look at that possibility as well.
I think when you have finished you will find it is either a combination of the switch and solenoids or the solenoids alone is the cause of your Chevy’s problem.