For example, if your dependent variable is crime rate, even after controlling for a bunch of variables, you will find out that the higher the number of police officers, the higher the crime rate, ceteris parabus. Does that mean police officers cause crime? Intuition tells us it is the other way around -- higher crime "causes" more police officers to be deployed.

So my point is, even if they controlled for a lot of relevant variables (which I don't know, as I actually did not read their paper), common sense tells me that there is still a causality issue here.