First, is there any theory for random sampling being optimal?
Second, consider the following example. Suppose there are two balls in an urn. Their colors can be either white or red. So there are three states: two red, one red on white, and two white.
In random sampling, one randomly draws a ball, puts it back, and draws once again. In this case, she cannot perfectly learn the states.
In a non-random sampling, one randomly draws a ball and draws the remaining ball. In this case, she can perfectly learn the states. So it looks non-random sampling is better.
Edit: Maybe I confused "random" and "independent". In sampling without replacement, the result of the first draw is (conditionally) correlated with the second draw. In this sense, my question may be rephrased as whether and why (conditionally) independent sampling is good.