Correlation also assumes model of the data generating process, but you are correct in thinking that talking about causal links imposes even stronger assumption

Correlation also assumes model of the data generating process, but you are correct in thinking that talking about causal links imposes even stronger assumption on the model and data structure for making inference. And then further you have to take a very narrow and convenient interpretation of what causality means (e.g. can't be at the actual level of individual samples, can't manifest through cycles or loops in the variables etc), which is even more of a vexing philosophical question than even the thorny questions in classical statistical inference