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.