I used to have an '06 Tundra and the manual said to lock out OD when towing. Unless I was towing a light trailer, like my 12ft aluminum boat, I would lock it out. If I towed my 8ft utility trailer (1,000lbs empty) I left it in OD with trailer empty. With a load of firewood, I locked it out.
My '06 Duramax/Allison has zero problems towing my 12K 5th wheel in OD, but that is a different animal. On a few occasions, I will go to manual mode and lock it to a specific top gear, but that is usually only in big rolling hills. In CA towing is limited to 55 MPH and I usually do 60. Mine does not even go into 6th (5th and 6th are OD gears) until 65 MPH. But at 60 it does not downshift unless it's a steep/long hill.
Having said that the Duramax and the Tundra are on opposite side of the towing world.