I don't know how good health care is in Italy. It could suck. But the Mayo Clinic couldn't cope if people started showing up there nonstop.
People have no idea how little excess capacity there is in ICUs anywhere. We have about 100 k ICU beds in the US and they are swamped during flu season. Patients routinely held in the ER waiting for a bed.
It may be true that any system can be overwhelmed, but the article says that the public hospitals asked the private hospitals for help, so they evidentally have some band width. It looks like Italy has a 2-tiered system where most people have to use the public option but the elite can go elsewhere.