"Wider tires .... all other things being equal."
OK, here's the problem with that statement: ALL other things CAN'T be equal.
If you go wider and don't change the aspect ratio nor the rim diameter, you go up in diameter and load carrying capacity.
If you try to keep the diameter and the load carrying capacity the same (or at least within a small difference), the aspect ratio and/or the rim diameter has to change.
Those 2 things can't be escaped from. So if you are trying to make a point about "wider", you need to indicate which of those is being defined as "wider".
So allow me to demonstrate: Wider tires have a larger footprint, all other things being equal (as best it can be done) - and it doesn't matter if "wider" = a larger diameter, or "wider" = same diameter, but lower aspect ratio and/or larger rim diameter. Both of those result in a larger contact patch.