Hi Jacob. We (@JHU) read your paper on problems with ML recently!
"On the other hand, some people take robust statistical correlation to be the definition of a causal relationship, and thus do consider causal and counterfactual reasoning to be the same thing."
These people would be wrong, because if A <- U -> B, A and B are robustly correlated (due to a spurious association via U), but intuitively we would not call this association causal. Example: A and B are eye color of siblings. Pretty stable, but not causal.
However, I don't see how the former part of the above sentence implies the part after "thus."
Causal and counterfactual reasoning intersect, but neither is a subset of the other. An example of counterfactual reasoning I do that isn't causal is missing data. An example of causal reasoning that isn't counterfactual is stuff Phil Dawid does.
If you are worried about robustness to model misspecification, you may be interested in reading about multiply robust methods based on theory of semi-parametric statistical models and influence functions. My friends and I have some papers on this. Here is the original paper (first author at JHU now) showing double ("two choose one") robustness in a missing data context:
Here is a paper on mediation analysis I was involved in that gets "three choose two" robustness:
I don't know how counterfactuals get you around model misspecification. My take is, counterfactuals are something that might be of primary interest sometimes, in which case model specification is one issue you have to worry about.