Your obvious indication that angle by sun > stars, then the answer will surprise you, it is not what you are thinking....
Lets consider Earth to be a point...and now its basic geometry that angle subtended by the stars is much higher then that of by Sun ?
The better thing for you would be to list down the voids in our observable universe, and calculate the angle from each galaxy or do modelling on those lines....even the angle subtended by our Milky Way is much higher ? This explanation is shot down.
It is a fact that lux is of the order of 0.001 (measured / observed aspect), how do you prove it.