No. There are plenty of random variables without non-uniform distributions. Plus an event in probability theory is not what you think it is. An event can be any subset of the sample space.
I don't think I'm being unrealistic, how can learning more about probability theory change the conventional definition of random?
Isn't the real difficulty in trying to make a non-deterministic system out of a computer, since all we ever do with computers is feed instructions into them?