statistical correlation is what Bayes is all about
No, I don't think so at all.
Bayes is about updating your estimates on the basis of new data points. You are not required to be stupid about it.
At a cursory glance, that site you linked does not appear to give any information on how it's generating those correlations, but the term "spurious correlation" actually has a specific meaning. Essentially, one can make even statistically uncorrelated variables appear to be correlated by introducing a third variable and taking the respective ratios and finding those to be correlated instead. It should go without saying that you should make sure your correlations are actual correlations rather than mere artifacts of your analysis method. As it is,...
Another month, another rationality quotes thread. The rules are: