I will agree that when acending to the rarified air that is The 40"+ Club, an 11.5"-wide tire may look ridiculously thin. Almost too narrow, but they work! Just look at our southern states' mud-whompin' events. VERY tall tires that are very, very narrow. Extreme case, but there it is.
But's that's not what this thread's topic was at the time of the OP. It was the topic of moving up to a 38" diameter, where an 11.5" section width is on the cutting edge (pardon the pun) of common tire proportion. a 38 X 11.5 is somewhat avant garde. It is a bold move.
But in the realm of 37" or 38" tire diameters, 11.5 (for a 37) and then 12.5 (for a 38") are my width limits. I stand by my assertion that especially when off road, a narrower tire does everything better. And, they're just easier on the truck.
As for the meme I posted, I must offer a correction to whomever it was that first created it: When airing down a tire it does NOT to make it wider to any significant degree, it creates a LONGER contact patch. Either way, airing down can and does improve traction.
What a wider tire will do best is when it's smooth or bald and that is to float on top of soft sand (as much as the vehicle's weight may allow it to float). At the same time, however, a wider tire will create on soft surfaces a larger "bow wave" of material in front of the tire's path. A narrower tire will have less to deal with when rolling through that terrain surface.
And a narrower tire will have a greater psi contact patch, giving greater traction all else being equal. The larger diameter helps, too, and for easier rolling over smaller obstacles.
And so on and so forth.