why do we need the other continuous distributions if everything is just converging to normal distributions

Question

today my teacher asked why do we need the other continuous distributions if everything is just converging to normal distributions when n is greater or equal to 30. I may partly understand the question but I want to understand it well.

Answer 1

Assuming you're referring to Central Limit Theorem, note that it has conditions as well (i.e. finite mean and variance). For example, sum/mean of Cauchy distributed RVs doesn't converge to normal distribution.

Also, the following question that's posed by your teacher needs context. Other distributions have very well use cases, too.

why do we need the other continuous distributions

Answer 2

Your teacher appears to be alluding a common misinterpretation of the central limit theorem.

The central limit theorem concerns the convergence of $ \dfrac{\bar{X} - \mu}{\sigma/\sqrt{n}} $, not of the data themselves.

Under mild conditions that are likely to be met in an introductory statistics class, as the sample size $n$ gets large (maybe $30$ is enough, maybe you need $30$ trillion bazillion), the sample distribution converges to the population distribution. This is the Glivenko–Cantelli theorem.

In other words, by Glivenko–Cantelli, as we get a larger and larger sample size, we get a better and better representation of the population.