I worked as an EMT and would regularly get bizarre calls from people abusing the system to take their kid to the hospital because he hurt his arm, the parents following in tow by car. Or a domestic dispute where they think it's reasonable to call an ambulance like it's a less-serious police dispatch to handle more minor issues. Or really just a free taxi service in general. -- You don't have to pay if you just walk out of the hospital or ignore the collections. If you don't have insurance, it's pretty much the only option.
I'm not sure if there are any alternative medical models that solve the nature of some people to abuse shared resources.
Nobody has infinite access because the access is limited by human intervention. It's limited based on need, not available cash.