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Causal calculus, mutual information, random forest importance scores, various hypothesis tests and other methods can all imply causation as well or better (especially in the case of non-linear or multivariate association) then correlation. All these methods and more are widely used in literature.

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Those methods still rely on correlation. To be clear: We are not only talking about linear correlation here, which is only one of several kinds of correlations.

No they don't. They work directly with underlying estimates of probability distributions, entropy or impurity decrease in machine learning models.

Another example: mendelian inheritance patterns in a pedigree study.

If you know of a good measure of non linear correlation please let me know. And publish a paper in science or nature like the MIC/MINE people did (a measure that has issues in practice).

To estimate probability distributions, you need data that is non-random. Non-random means there's a pattern. That's another word for correlation.

In using those methods you may never calculate correlation as a number, but when those methods find something you still rely on the fact that there is a correlation.

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