"If you take ordered pairs where there is no connection between a and b, then you won't get any regression to the mean, you'll get points essentially randomly placed on the plane."

This is exactly reversed. If A and B are perfectly correlated, then you will have no regression to the mean. If they are perfectly independent, then you will have full regression to the mean. If they are only partially correlated, then you will have only partial regression to the mean.

(This is easy to see if you run a simulation of each case.)