[matzf]

This already has the same problems as #2, but worse;this already has uneven distribution from the beginning, even without deletions. Perhaps it may be worth pointing this out: you mention that this creates a "perfect random distribution" but that "with only a few records the distribution is not as perfect". This is a misconception; this perceived imperfection, the clumps etc, that's what random points always look like. The size of the voronoi cells, which determine the probability of a point to be sampled, are of very different sizes here. What you are thinking of is a random distribution of points where the points will have somewhat uniform distance to their neighbors. This is called blue noise. Creating scattered points with blue noise properties in 2d is not very complicated, but it is a bit more complicated than just combining two uniform random values. FWIW, uniform randomness does not even have blue noise properties in 1d, perhaps much more obviously.

Anyway, if you had this blue noise, whether 1d or 2d, would still not solve your problem; once you start deleting points, you lose your beautiful properties of uniform voronoi cell sizes and your back to square one.