Fast Blur with Animated Radius

Almost a year ago, while testing some Blender compositor related things, I noticed that their “Fast Blur” mode is not very fast in some cases, and I started to wonder maybe Blender should get a Dual Kawase blur mode, but that had limitations, and after a whole bunch of poofing around I assembled a bunch of existing blurry ideas into Smol Gaussian.
So! As everyone knows, doing a Gaussian blur in the most naïve way is a cost that grows up as square of the blur radius.