Haha, if you think this doesn't happen in America just look no further then America's number one maritime employer MSC.
If you get sick while overseas you have to use your health insurance, MSC doesn't cover you. Everyone is required to have a credit card that's accepted overseas in case the overseas medical doesn't accept your health insurance. If you're truly in a bind, say the medical provider doesn't take credit then MSC can, in some cases, provide a loan if the need is dire.
Basically, you pay and your insurer reimburses you. I understand it's a law regarding federal employees that causes this.
A few years ago (2010?) we were in Dubai and a mariner fell at the gangway. He was shitface drunk coming back from liberty. He made it up the gangway but fell face first at the quarterdeck dislocating his left forearm at the elbow. (It looked like a closed compound fracture with a bone poking up from below the skin.) I was on watch standing two feet from him when it happened.
USN security showed up within minutes to take him to the hospital but our own medical officer wouldn't let him go. He needed a credit card. My watch partner ran to his room and had a hell of a time finding it. He was too drunk and in pain to explain accurately where in his room he hid his valuables.
After twenty more minutes she found his credit card and the USN security was finally permitted to take him to the hospital. So it happens and not just on foreign flags of convince.