Even though the surface area of a cone is $(\pi R G + \pi R^2)$, it makes sense to think that it actually is $\pi R\times \text{Height} + \pi R^2$ as you could think that the side area $=$ average circumference $\times$ height instead of being average circumference $\times$ generatrix ??? Is there something I'm missing?
R $:=$ Radius of the base and
G $:=$ The cone's generatrix
The formula (avg circumference)x(height) does not account for the slantedness of the surface.
Here's how to interpret your formula. While the cone is the surface of rotation for a diagonal line, suppose we instead rotated a "staircase" function (as in a Riemann sum for the diagonal line) around the $x$-axis, and we thus get a bunch of disjoint thin cylinders (say, inside) the cone. As the number of pieces tends to infinity and their sizes tend to zero, the total surface area tends to (avg circumference)xheight.
Consider the lower-dimensional version of this: does the staircase function do good as an approximation for the length of a diagonal line? Of course not; it fails to account for the slantedness.
Imagine on the other hand cutting up the actual cone into pieces. Each piece's surface area can be approximated by a flat, non-slanted thin cylindrical piece with the same generatrix length; we can get an over-approximation using the larger circumference and an under-approximation using the smaller circumference of the conical piece. Putting all the over/under approximations together gives nice bounds, and then taking the limit gives (avg circumference)x(generatrix).