I believe it is because electricity is fundamentally perfect in its uniform motion providing us to operate quantitative changes of its physical coordination when it is patterned periodically, permitting the extension of imaginary space to physical space via the reliance on the pure numerical relation between experimental measure and The Frequency Domain.

It is truly a godsend.

None of which I ever used in the software industry, even as a machine learning engineer.

I'd love to hear from someone who's used a Laplace- or z-transform professionally, for any type of engineering. What was your application?

Is it used in neural networks as approximators? From my understanding they're quite good but they suffer from overfitting.

Laplace Transforms are used in circuit analysis. https://web.stanford.edu/~boyd/ee102/laplace_ckts.pdf

What do you do with it; check for stability, like a unit test?

For a brief period i did work on audio, video, and image processing along with signal processing.

Laplace and z-transform are widely used in control systems and signal processing. Both are more general forms of the Fourier transform in continuous and discrete time, respectively.