well let's just say your privileges work on multiple levels
, imagine if you'd have chosen to live greece (I know, highly hyperthetical as their health service is in tatters due to reasons we needn't go into here). After registering with the municiple authorities as a resident (legal requirement) you then try to find a job but to no avail. Your savings dwindle away due to the costs of living and (long after 3 months presumably) you suddenly find yourself skint, on the streets of athens with no health insurance. At that point your're actually an "illeagal citizen" in the eyes of the authorities. As soon as you get bad dose of the lurgy you'd have nowhere else to go but back to your home country to get your statutory health insurance. At that point, you could theoretically rinse repeat (in other EU countries) in 3 month max cycles.