The normal distribution can approximate many other distributions, though the details such as quantitative error estimates and what factors improve or degrade the approximation are harder to find. Here are some notes on normal approximations to several common probability distributions.

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How bad does it get when the derived distribution uses normal values that in turn have been derived from uniform values, such as via Box-Muller?

And how good are commonly available uniform sources?

Long ago I wrote a radiation source simulator (and various radiation detector simulators) that relied on normal values generated using Box-Muller (Ziggurat didn’t exist back then). The output very closely matched that of the physical system, which permitted me to debug nuclear reactor instrumentation software in a much safer environment.

I think you’d need an astronomically large sample to reject the hypothesis that the output of a pseudorandom number generator is the real deal, provided you start with a good uniform generator such as the Mersenne Twister.

The line

“The error curve for ν = 30 is below.”

appears in the post relative to the beta distribution, but it probably only refers to the Student-t?