What is the weak link for Chevy max tow ratings?

I only drove one, an 05 my dad had. It was a 2005 Duramax 2500HD, 4x4, extended cab, short bed.

The temp gauge would hit 3/4 and the cooling fan would scream on any little grade when pulling his TT in the summer. His TT was a 26' Thor rear bath with 14" wheels. Not heavy or high profile, and no where near the max capacity of the truck, likely not even 1/2 it's max tow rating.