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.

A non-random sample may be good for a particular purpose, or it may be bad. A random sample can be shown with high probability to be "good" for many purposes.
In particular, in statistics, our purpose is to learn general properties of a population. There are some non-random samples we can draw that help us do that very well, but also some that mislead us. Without prior knowledge about what kind of sample is useful, it's easy to mess up.
On the other hand, when we draw a random sample, we can prove that with high probability, we will get a sample that is "good" for our purposes. Sure, there might be deterministic ways to get a good sample too, but those may require a lot of knowledge and planning. Random sampling is much easier.
In your examples, the sample sizes are too small for us to prove the good things I mentioned above. But if the urn has 100 or 1000 balls, random sampling will give good results.
