Here in West Texas where it gets over 100F for 3 consecutive months in a row ..... Just a bad idea to be running around with a poorly maintained cooling system.
When I first bought my old 1991 truck, I pulled and flushed the radiator both directions and cleaned it outside also.
I installed a new water pump, T-stat, all mew hoses and a serpentine belt .... Is simply called maintenance when you have no idea what was done to it in the past ..... While it takes a little time to complete, is pretty cheap compared to a flatbed tow home or a burned up engine.
Or sitting in the desert with no shade for hours waiting for a tow truck to arrive.
Tall 35" tires will make the truck work harder at a dead stop, but going down the highway your engine is actually working less ....
I use to run 33's on a old 65 chebby .... tires were not a issue and 35" on a modern truck ..... I say your tires have nothing to do with it.
Your cooling system is pooched and it is best to just go through all of it.