I do some light duty towing with my 05 Tahoe with the 5.3. works great when towing in 3rd and tow mode I wouldn't be against pushing heavier with it if i had heavier. but I would be and am concerned about pushing the 4l60e.
Our work truck is a 04 1500 with the 5.3/4l60e and that thing tows a trash trailer once a week that has a combined weight usually around 13,000 lbs. It only goes about 10 miles down the road loaded but it does handle it well especially up the hill it hits every week pulling.
However, I specifically bought a 2500 with the bbc/4l80e and bigger brakes for towing because I don't want to push my daily.