> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative

the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values

     f'(n) = f(n+1) - f(n)

Then the fundamental theorem is just a telescopic sum:

     f(b) - f(a) = \sum_a^b f'(n)