Good answers and relevant reference are already given. However it seem me that there is at least another useful point of view for clarify the problem. The idea to infer causal effects from conditioning is not new, It come from a lot years ago. Infact in some old references (maybe not only) the Bayes Theorem, the most relevant conditioning rule, was used also for try to discover the more probable cause of some effect.
Unfortunately this is not possible, Judea Pearl wrote a lot about that. We can read among others: Bayesianism and Causality, or, why I am only half-bayesian - (2001); but also the popular book: The Book of Why spent several page about the failed attempt to infer causality from conditioning.
The core of the problem is that, even if the conditioning sound like "what if" and, then, causal, the matter of fact is that conditional probability/distribution, in its proper nature, deal with passive observation while in order to achieve causal effects we need of interventions. Basically the problems explained in the other answers disappear if well managed interventions come in place. This discussion can give an idea: conditional and interventional expectation
Warning: the idea to infer causal effect from conditioning is not dead. Infact the experimental paradigm frequently used in econometrics as ideal benchmark (as if rule) make a wide use of conditioning as main tool; without ad hoc other mathematical tools. The trick is that if we add some causal assumptions to the story, the conditioning can be enough in order to deal with causality. However Pearl insisted al lot on the necessity of new and ad hoc language in order to "free the field" from ambiguity and strong limitations. I think that the future will prove him right.