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If you want to know how quickly the distribution of the sample mean converges to a Gaussian distribution, you can use the Berry-Esseen theorem:
https://en.wikipedia.org/wiki/Berry–Esseen_theorem
This highlights, for instance, that the most important factor for speed of convergence is the skewness of the population distribution.
I don't think it's fair to blame the mathematical statisticians. Any mathematical statistician worth his / her salt knows that the Central Limit Theorem applies to the sample mean of a collection of independent and identically distributed random variables, not to the random variables themselves. This, and the fact that the t-statistic converges in distribution to the normal distribution as the sample size increases, is the reason we apply any of this normal theory at all.
Press's comment applies more to those who use the statistics blindly, without understanding the underlying theory. Which, admittedly, can be blamed on those same mathematical statisticians who are teaching this very deep theory to undergraduates in an intro statistics class with a lot of (necessary at that level) hand-waving. If the statistics user doesn't understand that a random variable is a measurable function from its sample space to the real line, then he/she is unlikely to appreciate the finer points of the Central Limit Theorem. But that's because mathematical statistics is hard (i.e. requires non-trivial amounts of work to really grasp), not because the mathematical statisticians have done a disservice to science.