Taylor approximations converge slowly in terms of error if the function they're representing is discontinuous (the error disappears quadratically if continuous, linearly if not), and they tend to create highly energetic swings near discontinuties (similarly to Fourier series with Gibbs oscillations).
Moreover, Taylor series are inherently nonlinear, and much of the mathematical toolset around AI assumes general linearity (cue linear algebra), with the exception of sigmoids , and going beyond cubic approximations tends to make errors worse (as expressed in SNR).