That's not true. Look up doubly robust estimators for a neat counterexample. Or IPTW for an extension of a dataset that can help with this (X, t, y, and propensities, instead of just X, t, y). Causal inference has a super rich literature that's quickly growing.

Or better yet (stretching and sidestepping how you meant it), give your friendly neighborhood statistician/econometrician just a dataset and they can't do causal inference. Give them propensities and column names/descriptions and a writeup of the experiment/where the data came from, and suddenly they might be able to do causal inference. It points to a need to augment our observations with more structured metadata, if we want to do causal inference with data that's just lying around.