Here's why I say that proving the $4\pi r^2$ formula is more or less the same as proving Archimedes's theorem: The theorem is clearly approximately true for a very thin slice symmetric around the equator. As you move from the equator to a pole, taking slices of equal width, it would be at least mildly suprising if the corresponding spherical surface areas did not change monotonically. If they decrease, then the total area of the sphere is less than that of the cylinder; if they increase, then the total area of the sphere is more than that of the cylinder. So (at least given the expectation of monotonicity), having them all equal (i.e. the full strength of Archimedes's theorm) is equivalent to the sphere and the cylinder having the same surface areaareas as the cylinder, which in turn is clearly $4\pi r^2$.
Here's why I say that proving the $4\pi r^2$ formula is more or less the same as proving Archimedes's theorem: The theorem is clearly approximately true for a very thin slice symmetric around the equator. As you move from the equator to a pole, taking slices of equal width, it would be at least mildly suprising if the corresponding spherical surface areas did not change monotonically. If they decrease, then the total area of the sphere is less than that of the cylinder; if they increase, then the total area of the sphere is more than that of the cylinder. So (at least given the expectation of monotonicity), having them all equal (i.e. the full strength of Archimedes's theorm) is equivalent to the sphere and the cylinder having the same surface area, which in turn is clearly $4\pi r^2$.