Generally, yes. the 18 vs 20" deal is not real though. Plenty of 20s with 3500lb+ capacity.
You are correct about tires being available in the aftermarket, but not in the stock profile. Check it for yourself - GM only offers 18" wheels on the 3500 SRW trucks. If you check the stickers you will see that the RAWR is more than twice the load range for the stock 20" size. Disclaimer - this may have changed in the last couple of years, my info is up to date as of ~2017.
The stock size 18" tires are Goodyear SR-A LT265/70R18/E rated for 3527lbs.
The stock size 20" tires are Goodyear SR-A LT265/60R20/E rated for 3197lbs.
RAWR on the 2500HD is 6200lbs.
RAWR on the 3500HD is 7050lbs. 2x 3197lbs = 6395lbs, so the 20s do not work for the application.
Sure, they could have increased the size from the factory but the 18s and 20s have the same width and diameter (32.6 vs 32.5", or close enough, anyway), and thus can run the same running gear for all their SAE testing (plus not having multiple gear ratios to account for tire sizes, as all of the D/A rear ends were 3.73s in those model years).
Just increasing the tire size on the 20" wheel doesn't necessarily mean you are increasing the carrying capacity, either, since GM only rates the factory wheels for the OE tire load range. You will find a lot of 20" aftermarket wheels listed with ratings in the 3200lb range, but there are some that exist up to 3840lbs (last I checked). I used to run Method MR305 NV HD wheels with a 4500lb/wheel rating in 18" with oversized tires for load capacity. Also ran 19.5" Vision 81s for a while.
For whatever it's worth, I should have just bought a DRW years ago instead of fussing over wheel/tire and suspension modifications.