I understand what your manual says but I note no difference in mileage towing with or without T/H. I do notice a difference in performance however and use T/H when going up, down or against a headwind.
My 2015 Tahoe manual says: “The Tow/Haul Mode adjusts the transmission shift pattern to reduce shift cycling. This provides increased performance, vehicle control, and enhanced transmission and engine cooling when driving down steep hills or mountain grades, towing, or hauling heavy loads.”
You do as you wish. I’ll use T/H.