Not necessarily so. When you get to a too-thick bit, it puts undue pressure on the tongue and even the palate if the horse has a small mouth. It will be uncomfortable even at rest. Thumb width is honestly too wide for the majority of horses to easily carry. Pinky width is more reasonable, and is by no means thin enough to be harsh.
I found some pics.... the bit I am using has a sweetiron snaffle bit exactly like the fist pic, but it is shanked like the second bit. I do not yet clip my reins to the low ring on the shank, but on the middle ring as if it was not shanked. Just havent used a snaffle that thin before and wanted to know the difference in lay terms, between it and my d-ring snaffle, which is much thicker in the mouthpiece than the shanked bit.
Argentine snaffle, misnamed, of course, because it's not a true snaffle at all. Due to the size of the "Pelham" rings, I wouldn't really recommend attaching the reins there for long periods of time. It doesn't provide the greatest cue. But the width of mouthpiece is pretty typical (and not harsh).