Automatic differentiation only works for the simplest functions for which you already know what the Taylor series looks like. For those cases, you might as well just hardcode derivative functions and the basic derivative rules (linearity and chain rule). It is not a general-purpose method.
For functions that you can't even express by a simple formula, you still have to rely on finite differencing.
Don't call "legacy" anything that you don't understand.
Automatic differentiation works for any composition of those ‘simplest functions’ you mention, which is quite a lot of stuff, including whole programs.
Approximation methods have their place, sure. Sometimes they're good enough and sometimes it's all you can do. What does that have to do with anything?
If so, I look forward to learning about it. If not, you were throwing stones in a glass house.
There are also functions that you only know by sampling (e.g. ocean temperatures) for which you assume smoothness. You need to pick an interpolation method, but sometimes you do not interpolate beyond the sampling points, because that's just making up numbers. When you're limited by your original sampling step size, you have little recurse but to compute derivatives by some finite differencing scheme.