Where's the error in my reasoning?
We want to take an infinitely thin cylinder whose area is $2 \pi r$, where $r$ is the radius of a cylinder. Then $\int_0^R2\pi r^2 dy$ appears to be reasonable. However, to get the final answer we need to express how $r$ depends on $y$. We observe that $y=r$ and write: $\int_0^R2\pi y^2 dy=\frac{2\pi R^3}{3}$. Almost... But something is missing - namely, the factor of $2$. Where does it come from?
There are a couple of errors here.
First, the area of the infinitesimally thin cylinder is $\pi r^2$ not $2\pi r$ (which is the circumference).
Second, the relationship between $y$ and $r$ is $r = \sqrt{R^2-y^2}$ not $r = y$. You can see that this is the case by drawing a picture and using the Pythagorean theorem on the right triangle whose sides are a vertical segment from the center of the sphere to the center of the cylinder, the radius of the cylinder, and a radius of the sphere.
Finally, you need to integrate from $y = -R$ to $y = R$ to get the volume of the entire sphere. Otherwise, you just get the volume of a hemisphere.