The normal distribution has many properties that makes it easy to work with. It’s a handy shortcut to assume the distribution you’re working with is normal, and CLT is a good justification for why that’s reasonable.

The ingredient you might be missing for how common the CLT is applicable is the following: in most complex systems (e.g. biological systems), any variable you measure is likely influenced by a lot of other hidden variables. Because there are so many variables at play, each effect is likely to be small, and the way their effects are compounded is likely somewhat additive (this one comes from things like series expansion). Hence summing up effects between variables must be relatively common and account for the bulk of the variation in a response variable.

A last bit is this: most statistical methods like the linear model are relatively robust to deviation from the normal distribution. So, you don't need exactly a normal distribution, you just need close enough. It turns out the CLT often produces "close enough" quite quickly (i.e. with a few variables added together).

A distribution is gaussian until proven otherwise. Is a great shortcut into statistic. It's also why something as simple as the Chi^2 test works very often. No need to tune a complex likelyhood function or set a neural network, you can use a simple method which in many case is good enough

3blue1brown have a series on that.

Afaiu, it's because it's simplify a lot of statistics. If you don't know the underlying distribution of the population you are studying, you can be sure that taking a lot of samples and taking the mean of that you are going to get a normal with a mean that is the same as the population.

Is in undergraduate topic? 😰