in that a counterfactual x' with the exact same features but different S would be classified the exact same way.

Except that from the x, you can often deduce S. Suppose S is race (which seems to be what people care about in this situation) while X doesn't include race but does include, eg, race of parents.

And I'm not aiming for S uncorrelated with Y (that's what the paper's authors seem to want). I'm aiming for S uncorrelated with Y, once we take into account a small number of allowable variables T (eg income).