That has probably more to do with Poland being overall piss poor compared to the countries that always rank top on health care things (like Norway) rather than how you're paying for it.
The big difference with private - you can choose how much do you pay and what you'll get. With public you get poor experience no matter how much you pay. And sum is not limited: more you earn more you pay :(
I'm not sure what kind of health insurance I could buy in the US that would give me short queues. Lower deductible, more ancillary services, sure, but what's the plan that says "no more than a week to schedule any kind of appointment you need"?