You would fix the seed once in the beginning before sampling.
Also you would need to sample exactly N-times as there no rejection required. as the points will all be within the circle by construction and will have uniform distribution
The article was specifically asking for concentric rings with equally spaced points - how would you fix the seed such that this property is held?
The problem was stated in the beginning as 'Consider the problem of generating N points on a disk of diameter D. The function must be deterministic and the points somewhat uniformly spread around on the disk. Importantly, I want to generate exactly N points, for any integer value of N.' Then the author decided to do it through concentric rings.