You can get into similar difficulties in just two dimensions.
Consider an isosceles right triangle with legs of length $s.$
If you place one leg horizontal and one vertical,
you can slice the triangle by closely-spaced horizontal lines
and put as large a rectangle as you can between each pair of lines
so that the rectangles nearly fill the triangle.
As you reduce the spacing between the lines toward zero,
the total area of the rectangles approaches $\frac12s^2,$
which is the correct area of the triangle.
But if you try to measure the length of the hypotenuse by adding up the
short edges of the rectangles that are along the hypotenuse,
you will get a total length $s,$ whereas the correct answer is $s\sqrt2.$
The reason the rectangles work for area but not for length
is that the exact shape of each horizontal slice is a trapezoid,
and the rectangle gives a good approximation of the area of the trapezoid
but not a good approximation of the length of the slanted side.
When we fill the triangle with thin rectangles, the "missing" parts
(the parts of the trapezoids that are outside the rectangles)
form a sequence of small triangles along the hypotenuse of the large triangle.
As we slice the triangle into greater numbers of slices,
the total area of all these triangles goes to zero,
but the hypotenuse of every triangle is still $\sqrt2$ times its height,
and adding up the heights will always give you $1/\sqrt2$ times the actual length of the hypotenuse.
In your cone, when you fill it with cylindrical slices, the total volume in the "missing" parts of the frustums gets smaller and smaller as the slices get thinner,
but the lateral surface area of each frustum is nearly $S/H$ times that of the cylinder inside it (for most of the frustums)
and never less than the lateral area of the cylinder,
so the total lateral area of the cylinders never approaches the area of the cone.