Did you even read what you just posted? I'd have to pay out of pocket if I'm not a legal resident or have an international health care insurance. For the love of Jesus Christ, don't be so shameless. Read your own sources.
"Private hospitals normally have higher fees than public hospitals. Patients in private hospitals are generally required to pay fees upfront before leaving the hospital. (Patients with insurance may subsequently seek reimbursement from their insurance companies.) Some private hospitals may be “accredited” which means that fees can be reimbursed by the Italian National Health Service if the patient is an official resident in Italy and affiliated with the National Health System. It is highly recommended that tourists obtain medical insurance with overseas coverage before travelling to Italy.
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insurance providers for overseas coverage."