Is it possible to "Blur a Transparent Background" in Adobe After Effects?
It's not possible. A generic blur effect relies on a function which smooths out pixel values within a specified radius. A mask combines the value of a pixel in the top layer with the pixel of the layer beneath, modulated by the corresponding alpha value of that pixel in the top layer.

So, in short a blur is a 'lateral' function, whereas layer compositing is a 'depth' function.