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What is the interpretation of the do() operator?

I don't think one should see Pearl-type theories, which fall under the general heading of interventionist accounts, as reductive theories, i.e., as theories that reduce causal relations to something non-causal (even though Pearl might claim that his account is indeed reductive). I think such theories indeed make irreducible appeal to causal notions in explicating causal relations.

One reason why this isn't problematic is that these theories are explicating causal relations between some variables in terms of causal relations between those variables and the interventions and correlational information between the variables. So such theories are not employing causal information between the variables themselves in order to explain causal relations about them -- which would indeed be viciously circular. This point is explained clearly here.

If you want a reductive account of causation, I think that's a much harder problem, and indeed there might not even be one. See here for more details on attempts to provide reductive accounts of causation.