I think you are overstating Judea Pearl's argument. He doesn't claim it is impossible to answer causal questions (on observational data) without resorting to do-calculus. What he does claim is that to answer causal (interventional or counterfactual) questions you need to define a model and reason on that model in a way that transforms the question into conditional probabilities. Do-calculus provides a framework to help you perform that reasoning, but certainly people have been using things like the backdoor criterion before the invention of do-calculus to answer interventional questions.

He might claim that any other method can be reduced to do-calculus. I'm not sure. I do believe at the core of his argument is need for an explicit model.