Most of the friction or 'ease' of cutting through thicker material is due to the thickness and shape of the steel behind the edge, not the edge itself. The edge obviously needs to be sharp enough to initiate the cut (which is even more important for some things, like slicing a tomato or a grape), but once the body of the blade starts making contact with the material, that's where the 'ease' of cutting (or lack thereof) really becomes apparent.
The most obvious comparison would be in putting edges with identical geometry and refinement on two blades of different thickness, such as putting a 30 degree edge on each of a 1/16" thick blade and a 1/4" blade. Cut into some thicker material, or even slice into the edge of a piece of paper, and the difference will be obvious once the thicker blade is deep into the cut. The thicker blade will behave more like a wedge, in trying to split the material. That adds a lot of resistance in a deep cut.
Sometimes, such as when slicing a tomato or a grape, even a very thin blade might not help. I recently attempted to use a very thin kitchen knife (maybe 1/16" or less in thickness) to slice a tomato, and the edge itself was so blunt, it wouldn't cut through the tomato's skin. Ended up using my pocketknife, which was a thicker blade, but very much sharper at the cutting edge.
Another way to greatly improve ease of cutting with a thicker blade, is to convex the edge. Just rounding or smoothing the shoulders of the bevel can vastly reduce the friction normally seen with a thicker, V-bevelled edge. The hard-edged shoulders on a thick V-bevel can really create a lot of drag in thicker and tougher material, like cardboard or leather.