"Shade Smooth" does not seem to work in the way I expected

In the screenshot below, I created two cones, and I applied “Shade Smooth” to the right one. The thing I expected was smoothing the gradients that I see in the left one, but if you see the right one, the shading is less realistic than the left one. There is a round lighted area as if the cone is a sphere. The left one, although it as gradients, is more physically accurate, in my opinion.

So, I wonder, is there any way to make “shade smooth” as I expected: removing gradients but physically accurate?

Try auto-smooth control:
https://docs.blender.org/manual/en/latest/modeling/meshes/structure.html#auto-smooth

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Shade smooth has nothing to do with materials, or shaders. There is no need for the viewport view to be physically accurate. All shade smooth does is smooth the geometry and remove the blockiness. If you want a nicer looking viewport view, you can turn on a matcap. Adding extra geometry will also generally make smooth shading look cleaner, though cones will always look a little odd by default due to their topology.