Why is the Cobb-Douglas production function so popular?
As you hint at in your question, the real underlying reason (in my opinion) behind the C-D production function's popularity is mathematical convenience. The fact that the sum $\alpha + \beta$ is a nice, intuitive representation of "returns to scale" is very convenient.

I think of its use as similar to the use of exponential or power utilities in mathematical finance. Unrealistic? Perhaps, but oh so friendlier to work with.