Neither statistical technique nor design induce causality in and of themselves
Take the surest of situations- the randomized experiment
Suppose an effect of the treatment is discovered, it may be that
Outliers are an issue, and in such a case the effect is with only a relative few
Even if not outliers, the treatment may not, and likely will not, affect everyone the same way, and while the statistical technique used assumes ‘general’ homogeneity, it does not make it a reality
The model may still be misspecified
The effect is there, but in fact the treatment causes a mediator, which then causes the outcome of interest
One can use any ‘soft’ modeling statistical technique to analyze experimental data
Using regression on experimental data does not detract from its causal conclusion