Towing with Chevy Suburban 1500

I've been towing since 2006 with GM's ubiquitous 5.3L, in my case always with a 3.42 axle which is not nearly as capable for towing as your 3.73. Unfortunately that 3.73 is coupled to the older wide ratio 4-spd transmission, which in turn means the only way you can realistically tow with this powertrain is to put it in 3rd with Tow/Haul engaged and just enjoy the ride. Yes, on serious upgrades it will slow noticeably and will if necessary downshift to second in order to maintain an adequate road speed. Having a 3.73 rather than a 3.42 does help but there's only so much that little 5.3 can do. The biggest improvement I've seen is when I recently replaced my 2005 Avalanche for towing with a 2012 Silverado with the newer narrow range 6-spd that handles towing loads much more intelligently than did the older 4-spd. Now I just engage Tow/Haul (which locks out 6th), put it in Drive, and let the transmission figure it out, most times running in 5th, occasionally dropping to 4th when headwinds or upgrades require it. In your case however it is what it is and your report to me sounds like just what I would expect towing a trailer of the size you have with your GM 5.3, equipped as it is.