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Or use a machine learning method that is fully invariant to all monotonic transformations, for instance random forest.

Quite true. And, sometimes things are already in comparable units. For example, grayscale image intensities.

I don't see the problem with PCA if it is used sensitively to identify the approximate dimensionality of "pancakes" embedded in a larger space.

Wrong.

Neural networks are affinity invariant. You can rotate, skew, translate the input data however you want, the optimum stays the same. Same for SVMs.

But the weights will be different, which it seems to me is comparable to saying that the principal components change.