This sounds like a rule of thumb.
This is not a rule of thumb. It is the mathematical formula for accelerated stall speed. It is no more a "rule of thumb" than A squared+B squared=C squared is a "rule of thumb" in relation to right triangles.
I do not believe the stall limit/aoa is the same for every wing. 
You are of course correct here, but that is irrelevant to the topic. Stall speed is simply the minimum airspeed required for the plane to produce enough lift to keep its own weight in the air. So on and so forth with accelerated stall speed.
Turn rate and radius for a plane at any given moment will be defined by Gs and airspeed, NOT by the critical AoA of the airfoil. Given the same airspeed and the same G load, any two planes will have the same rate and radius of turn. Being able to produce more G at lower airspeed is better. Best possible turn rate and radius is achieved at corner velocity, the lowest speed at which the airplane can pull maximum allowable Gs. The critical relationship is the one between the lift available to the airplane and the weight of the airplane.
To give you a practical example, many WWI airplanes have relatively bad airfoils that are little more than flat plates. This is because the science of aerodynamics was not as well understood when many of these planes were designed. Compared to the airfoil of say, a Hawker Hurricane, they are really inefficient. None the less, these WWI have a lower stall speed and will turn in a smaller circle than a Hurricane simply because the loading on the wings is so light.
The airplane that has a lower stall speed WILL have the better instantaneous turn rate and radius at corner velocity, assuming both aircraft are operating under the same G limitation. It is very likely to have a smaller sustained turn radius, because turn radius is so strongly a function of stall speed. Sustained turn rate is another matter, an airplane with a heavier wing loading and higher stall speed can sometimes still have a better sustained turn rate via having a lot of thrust and the ability to sustain more Gs in a turn.
