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.
Plenty of European countries give you the option of public or private insurance. I live in Spain and we have both. Even the NHS lets you buy private medical insurance. It's just that most people don't.