If you do this you won't get a uniform distribution on the circle, the points will be the most dense at the center and get less dense as you go towards the edge. To make the points uniform you need to use inverse transform sampling[0], which will give the formula r*sqrt(rand()) for radius poolcoordinate, where r is the radius of the circle and rand() returns and uniform random number from the interval 0 to 1.
Yes, though you have to take the square root of the sampled radius for the resulting distribution to be uniform on the unit circle. (The area of the donut with r>0.5 is greater than the area of the circle with r<0.5, but the naive implementation would sample from each of those with probability 0.5.)
It’s still a useful illustration, though, since MCMC samplers used in practice do end up throwing away lots of the sampled points based on predefined acceptance criteria.
