Should I tow a 30' 7" 5820lb trailer with 2005 Yukon Denali XL?

I see you are in Colorado, here in Michigan that set up would have plenty of power to tow that trailer, even if that is the empty weight. You are limited on payload. With a family of six and with 12-15% tongue weight you will exceed the rear axle weight rating and some of the other weight ratings.

A 2010 or newer GM van with the 6L/6speed would tow even better than your Denali. Some people prefer the Ford vans but you probably would want to look for a V10.