To address your first point – that's fair, I didn't really dive into the statistics in detail (since it wasn't really the point of the post). If variance of a distribution is finite then the central limit theorem applies, and given a sufficient number of trials, the distribution will begin to approach a normal distribution. However some data sets have infinite variance and may (under some conditions) begin to approach a power law distribution; or, they have finite but extremely large variance, in which case the number of trials it will take to begin to look like a normal distribution is very, very large.

For your second point – there are legitimate uses for other summary statistics (mode, median), but they can still be very misleading. For example, you mention using the mode of the distribution as a predictor of what the next customer will pay – this definitely wouldn't work for a company with a metered pricing plan, for example. Distributions are often not well characterized by singular summary statistics, Anscombe's Quartet (mentioned elsewhere in this comments section) is a good example of this: http://en.wikipedia.org/wiki/Anscombe's_quartet