I hope this doesn't sound too vague or like I'm dismissing the use of Taylor Series entirely, I'm just curious about any proper real-world applications.
Many times Taylor Series are shown-off as a tool to simplify complex functions down to just nicer polynomials, perhaps the most famous example of this is that of a pendulum, where $\sin \theta \approx \theta$ so we get $\ddot \theta = -\frac{g}{\ell} \theta$. However, this really is only valid for around $\pm 5^\circ$, which also means the formula for the time period is also only really valid around $\pm 5^\circ$.
Outside of exam questions, what's the actual realistic need for this?
In an age of computers, we can just numerically simulate the entire equation ($\ddot \theta = -\frac{g}{\ell} \sin \theta$) without any need for an approximation at lightning fast speeds allowing us to start at any initial value of $\theta, \dot \theta, \ddot \theta$ we want. We can even add a dampening term proportional $\dot \theta$ at virtually no extra cost. It's sort of what makes computers so great.
So, is there any scenario where a Taylor Series is critical for something to work, where a computer simply could not simulate numerically directly?
I want to stress I'm not asking about any form of linear approximation, just Taylor Series specifically.