Why is random sampling good? 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.

The Central Limit Theorem may be the theory you're looking for.  It shows that random sample means follow a Normal distribution (even if the population isn't Normally distributed) and that allows us to use a lot of popular statistics like standard deviations, p-values, etc.
Of course, if your entire population of interest is two individuals, then you take a census, not a sample.  Sampling is used when the population (of individuals, events, etc) is too large for you to collect data on every individual case, or when it's otherwise impossible to do so (e.g., some cases are future events).
