The problem with this mantra is that it is technically true, but, as a negative, leaves out the rest of the world. For example, Correlation also does not imply non-correlation. This is vacuous, of course, but it illustrates the problem with using this kind of shorthand to quickly dismiss something rather than investigate it.

In truth, correlation does suggest causation, but not necessarily in the way one might expect. The WP article you linked goes into this.

All too often, this canard is trotted out to avoid thinking about where the causation might actually lay, rather than taking something seriously. It is precisely this that the shorthand is meant to prevent! Correlation does not imply causation, therefore you have to think carefully about the causation chain when considering two correlated events -- you can't just assume the causation isn't elsewhere, and you most definitely cannot assume it does not exist at all!