Because for random variable X, sometimes need to care about X^2, and when X is normally distributed X^2 is chi-square distributed and not normally distributed.

Because for random variable X, sometimes need to care about X^2, and when X is normally distributed X^2 is chi-square distributed and not normally distributed. Because under mild assumptions, for arrivals, say, of visitors to a Web site, gas station, hospital, the arrivals form a Poisson process, the times between arrivals are independent and identically exponentially distributed, and that's not normally distributed. Now at nearly any server farm, it is easy to get wide, deep, rapidly flowing oceans of data on the performance of the server farm, and, as U. Grenander once explained to me in his office at Brown, the data is wildly different from what statistics was used to, e.g., in medical data. In that ocean of data, finding anything normally distributed will be very rare. The claims in the OP about many effects are nothing like good evidence for the central limit theorem or normally distributed. E.g., from the renewal theorem, many examples of Poisson processes are from the results of many independent effects. E.g., the usual computer based random number generators return what look like independent, identically distributed random variables uniform on [0,1], and that is not normally distributed. The question in the OP about why not normally distributed is, in one word, just absurd.