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).
How so?