I can attest, from years of towing our 28', 6-ton (loaded) toyhauler - with a Silverado 2500HD Duramax/Allison, the OP's info is 100% correct. I wish it had been out there when I was first starting towing.
I whole-heartedly recommend the Prodigy brake controller... it has performed flawlessly from the moment it was installed.
We chose to replace the OEM unit with a Putnam hitch, as it has a significantly higher rating than the Drawtite or Reese units (16,000/1600). The stock Chevy hitch at the time had a reputation for catastrophic failure (i.e. coming apart while towing).
The only other thing I would add is, take your fully loaded trailer to a certified CAT scale (truck stops often have them). For under $20 you will know your EXACT gross trailer weight and tongue weight. If it's a slow day, and you can spare a few bucks more, you can even get the weight on each axle of the trailer and truck. In Mav's bible, he says the weight of your trailer will surprise you, and get very heavy, very easily...I fully concur with that. In addition, trailer manufacturers often list the trailer weight BEFORE any options (like the genset, bbq grill, tv, etc.) are added. Water & fuel contribute significantly to the weight (our toyhauler holds 150 gallon of fresh water, and 50 each of black & gray... though they are almost never all full at the same time, that' a total of 2075 lbs!). All before loading our bikes...
WEIGH YOUR RIG!!!