It all depends on what you consider skinny sidewalls   You can buy a 35 inch tire that goes on a 20, 18 , 17 or 16 inch rim.  The tire is 35'' in diameter regardless of the rim size, But I wouldn't consider any of these rim sizes to be low profile and they are all in common use in todays trucks. A 35'' tire on a 16'' rim will have 2'' more of side wall above and below the rim than the 20'' set up.

Low profile tires don't have as high of load rating.  So they wouldn't be able to carry as much of a load.  Also their is less cushioning if you run over a curb or other object. Your chances of bending or damaging a rim are higher.

Low profile tires do offer less sway in the sidewalls under hard cornering, so their purpose was originally in the race cars. With the popularity of the low rider cars and those wanting a race look, their use has spread, But they still are not very practical for heavy loads.

  Just depends on the tire. An option tire on a 2011 F-250 is a LT275/65Rx20E, it's rated at 3750# at 80psi. Dual load rating is 3415#.  What was the size and load rating on the tires you seen on the truck at the show...??

I think the skinny tires he is talking about at the 35-40-45 aspect ratio tires.

Such as the

295/40R20 which is also on a 20" rim, it's load is only 2601 lbs

285/45R22 which is on a 22" rim, It's load is only 2601 lbs

or the 305/35R24 which is on a 24'' rim, It's load is onlyu 2469 lbs

These are all considered Low Profile tires for Larger SUVs and not really designed to carry a load.

Any of the tires in a 65/70/75 aspect ratio are considered truck tires and can handle the loads if rated in the appropriate load range.  The cavat here is the folks who buy floatation tires. The tires designated at 33x12.5 by X rim size. or the 35x12.5 by X rim size.  These tires while rated at 10 ply and E rated still run at 65 psi and their load ratings are considerable lower than the metric truck tires of similar size and laod capacity