I think the calculus-based explanation is the simplest one, and also easy to extend to an arbitrary number of dimensions.
If we look at the direction from the tip of the cone to the base, the volume is an integral of the area of the circles that are the cone sections. The radius of those circles grows linearly when the point is moving from the tip of the cone to its base. The area grows proportionally to a square of the distance from the tip. An integral of x^2 is x^3/3. Hence 1/3.
(x^3)/3 is so much faster for me to read and far harder to read incorrectly. Yes there's the memory trick for order of operations, but division symbols are so easy to get confused around that the extra clarity of the parenthesis should be standard for all non-trivial (one item per side) cases.
So then if we started with (x^3)/3 the 3s would cancel and we’d get x^2. This tells us that the antiderivative of x^2 is (x^3)/3.
Or did you mean some other intuition? Such as why the fundamental theorem of calculus (that integration and differentiation are inverses of one another) is true?
Graphing y=x^0 -> y=x^1 and y=x^1 -> y=(1/2)(x^2) and doing math on the shapes (just simple squares and rectangles) makes those pretty intuitive. I just see it as an extension of that.
If we look at the direction from the tip of the cone to the base, the volume is an integral of the area of the circles that are the cone sections. The radius of those circles grows linearly when the point is moving from the tip of the cone to its base. The area grows proportionally to a square of the distance from the tip. An integral of x^2 is x^3/3. Hence 1/3.