My guess is that the "sloppy" feel has more to do with the age and wear of the specific truck involved. Consider that the newest GMT-800 Suburban is now 12 years old and most of the ones you see on the used market are well used. When buying something this old it's realistic to consider replacing (or even upgrading) some of the wear components in the steering system. Mine is still pretty "tight", at least enough that I don't feel like spending the money to change anything. As far as a "sloppy" feel, honestly my 4runner was worse, not sure why. It seemed to wander a bit at highway speeds, especially with the trailer behind it. The Suburban seems rock steady, even at 172k miles.
The 5.3 in the GMT800 is not known for being extremely powerful and the 4 speed auto that puts a huge gap between 2nd and 3rd doesn't help much either. On paper, at least, the newer 5.3's in the GMT-900 trucks have significantly increased power. Of course, that power comes with increased complexity and more things to go wrong (like the infamous AFM or Active Fuel Management system that shuts down 4 cylinders for better MPG.)
Another factor is weight, a Suburban weighs almost 500lbs more than a 1st gen Sequoia.
Honestly, every choice you make in life is going to be some kind of compromise. A 12+ year old truck is going to have a lot of compromises built in, so ultimately it comes down to what you can live with. Between the size, the capability and the reasonably low cost to buy and maintain, many of us have decided that the GMT800 Suburban works for us, but that doesn't neccessarily mean it will work for you.
Much as I like the Suburban platform, I've decided my next vehicle will have to be a pickup. Not because I don't like the Suburban (I actually prefer the "single body" design of a big wagon vs. a pickup) but because GM, in their wisdom

has decided to remove most of the "utility" from the Suburban design and turn it into a big station wagon with plenty of perks for passengers but virtually no allowances for cargo or utility (the non-flat folding 2nd row and non-removable 3rd row, the removal of a low-range transfer case on 4x4 models except as an option, etc.) For that reason as well as cost I've come to the conclusion that a full size half ton crew cab will better suit my needs, even though that entails a lot of compromises too.
As an aside I still think it's a shame that GM decided to kill the Avalanche back in 2013. It was a genius idea to have a convertible cargo/passenger area. I just think not enough customers really appreciated it and ultimately just used it as a pickup without ever resorting to using the disappearing mid-gate for bigger cargo, and because of that, GM figured that the same customers would likely just buy a cheaper-to-make crew cab pickup instead.