wheels on towed vehical not following in a turn

Did you follow the instructions in the manual for towing it ? My GMC has explicit instructions to tow it.

Didja try leaving the engine running just for a test ride ?
If it doesn't steer w/ engine running, then do you KNOW that it is even designed to tow 4 down ? I'd bet it isn't.

My 2012 GMC Terrain is a four cylinder front wheel drive model with 20,000 mi. I purchased the vehicle 3 months ago for the purpose of towing behind my Four Winds motor home. I had a Blue-ox tow system installed at a reputable RV center and have been dealing with this issue ever since. I am about ready to give up on the vehicle.
The vehicle has been inspected several times by GM mechanics. They tell me that there is nothing wrong with the steering or the suspension. I keep telling the mechanic that there is something different about this car but no one has been able to help.

Is this coming out of a tight turn, like hard right at a side street, or even a U-turn? A possibility, a little hard to describe, is this, using a right turn:
When Coach turns Hard Right, its long Rear Overhang swings LEFT
Toad therefore turns LEFT
When Coach straightens out, Toad wheels are still LEFT
Toad wheels scuff sideways instead of returning straight

We towed tow small Jap vehicles, manual transmissions, manual steering, steering unlocked. One, a mini-pickup, always came out of a turn. The other, a FWD subcompact, sometimes did not. I'd never tow it with the doors locked, so I could bail out of the RV and go straighten the toad.

Is the hitch ball at the same level as the hitch at the vehicle? Substantial differences in height can cause such a problem.

Have someone ride in the car for a short distance (around the block). The tires on my tow do not turn much, sometimes I wonder if things are OK, but over 100K miles of towing things are good.
If you can see steering wheel of towed in rearview camera tie a rag around top center of steering wheel and monitor its movement.

are you pulling fuse #32 to disable the ign system and the electric power steering assist. You might want to try that

Since you've indicated the key is in the ignition and the steering wheel is unlocked, there aren't many other potential causes.

About the only solution I would suggest at this point is to have someone ride in the truck while it is being towed and have that person try and turn the wheel in the correct direction when you make a turn. If the truck's wheels turn in the direction your coach turned, then you'll know it is a "following" problem. That is, the suspension and steering system in the truck won't respond to being pulled around a corner. If that's the case, it is a basic design flaw that probably can't be changed.

On the other hand, the person in the truck may be able to tell right away what the problem is and it could be a simple thing like not leaving the key in the right position.

Sounds like the steering wheel is not being unlocked. It must be something in the steering column set-up.
What type of drive is in your rig? AWD, four wheel drive? What transfer case?
After inserting the key, does the steering wheel turn?