A confidence interval won't adjust the points (point estimates) but will give those points with a lower sample size wide confidence intervals (often covering zero).
Using an (empirical) Bayesian multilevel model can both attach uncertainty intervals to the point estimates and appropriately "shrink" the estimates towards zero at the low-sample-size end.
The latter is more directly interpretable, at the cost of slightly more complex modelling (/assumptions).
Just so people know, there is a competing/complementary approach to causality in statistics, called the potential outcomes or (Neyman-)Rubin causal model, which as I understand it is currently more popular than Pearl's graphical/do-calculus approach.
I don't think machine learning vs Bayes vs sampling theory has much to do with the content of the article, which is more about causality than interpretations of probability.
> it's far more likely that coffee causes cancer if it can accurately predict it, even when you control for all other variables
I don't know about this: prediction is not equivalent to explanation in general. The "all the other variables" bit is also a bit of a kicker (what counts as "all"?) -- hence randomization, and, well, pretty much everything else the article discusses.