Arguing that you don't care about the right to privacy because you have nothing to hide is no different than saying you don't care about free speech because you have nothing to say.
Then there is every second a nonzero chance that this machine (assuming true and not pseudo randomness) will pick, say pi.
No. The probability of picking any particular number from a uniform distribution is 0.
On the contrary, since the works of Shakespeare are a finite string over a finite alphabet (I can formalize this argument if you want), the probability of typing them out after some fixed large number of keystrokes is some nonzero number 𝑝. With 𝑛 monkeys, the probability that at least one will type out the works is 1 − (1 − 𝑝)ⁿ, which goes to 1 as 𝑛 → ∞.
Now, you are right that this does not mean that the works are guaranteed to be typed out. However, it has probability 1, so it’s mathematically “almost certain”.
This sample also shows why the median is often more representative than the mean.