Now, it may be that there is a different relationship, like the one that you mention: that eating more than X grams per day is harmful. The data from an observational study can negate this hypothesis (if there is no threshold pattern evident in the data), but it cannot prove the hypothesis. All it can do, at best, is let you know that the hypothesis might be true.
This is very rare and hard to run, so more direct interventional short term studies are used more often.
Suppose a certain country believed an observational study pointing that eating more than certain amount of sugar is harmful (suppose backed by other evidence). They instated hard tax and limit on sugar in foods as an intervention. After some time a big observational study is ran and it is assessed if the intervention brought results.
This is precisely what observational studies (follow up) should do. Not really try to find hypotheses - these need more direct evidence.
Any nonlinear enough process will show with wrong low correlations.
Suppose the process is stepwise - silly sample hypothesis, eating over x grams meat a day is harmful. It is binary. Correlation is meaningless for such a process. You have to use significantly more powerful statistic tools. Logistic regression will somewhat work for such a process for instance if the boundary is somewhat fuzzy. (You can put exact mathematical bound on how fuzzy.)
To extend it for multivariate analysis you get to use such complex tools as nonlinear ICA. (Instead of CCA and PCA for linear.)