[rrss]

From a certain point of view, modern deep neural networks for audio are 'just' nonlinear adaptive filters on steriods.

Linear adaptive filters have been around for a long long time, and nowadays are everywhere. They can't capture the nonlinear behavior of effect pedals, not even just the waveshaper.

The model you are describing sounds like a 'wiener model,' which refers to a linear filter followed by some nonlinearity (i.e. the waveshaper).

There are other approaches to nonlinear adaptive filters, like Volterra series and kernel methods.

People have been using all of these techniques, and more, to approximate analog audio effects for decades.

A 'trained deep neural network' is not in principle that much different or 'less pure' than other nonlinear adaptive filtering techniques, just with a load more parameters. What matters is if the results are sufficiently improved to justify the computation.