The best 3 ATs that I’ve tried are the Toyo Open Country AT3, Cooper AT3, and BFG AT KO3s. I have read a lot of positive reviews on the Falkens and would like to try them, but have no personal experience. I do remember that comparing the Falkens to the others in the same size, the Falkens weighed more, so it’s possible they have more beef which would help you with flats? Hard to say. I think they are on the AT4 version so that could’ve changed.
The BFGs seem to be the standard that ATs are judged by. Mine were on a 2500 Cummins and they were a good tire, but I was less than impressed with the tread life. Use was mostly highway, minimal towing and off-road. They were replaced at around 40k with the Coopers. The Coopers were cheaper and wore better. They were at 50k when I sold the truck. I’ve been most impressed with the Toyos and have had 3 sets between two Toyotas. They ride well, wear well, aren’t the most expensive, and are impressive off-road. Use case is highway miles, moderate towing, and consistent sand and mud off-road use. But, those vehicles aren’t 3/4 ton work trucks.