The normal distribution does indeed make some assumptions. Certain natural qualities tend to be notmally distributed like height, but other things like net worth are NOT normally distributed because of feedback loops. The choice of distribution either makes an assumption or validates some other prior assumptions.

Thr choice of using a normal distribution vs another mathematical distribution is not purely a mathematical device, the choice either reflects some assumptions (which one would quickly see if they're valid or not generally).

An interesting case in point for me is when a lot of mathematics assume bell curve distribution in stock market price distributions, like the black scholes option equation does, and it turns out to not really be the best fit. It sort off almost works, but systematically underprices extreme events.

This is what can happen when your assumed distribution turns out to be wrong.

So far though, I haven't heard of refutations that height or IQ (as measured on standardised tests and processed in prescribed ways) are indeed bell curve distributions.