Currently, their insurers are not heavily restricting them, because they're paying for the privilege.
But honestly I think single payer is just part of the solution. You also need public hospitals, because hospitals have no incentive to suggest a good but cheap solution to a medical problem.
I read a book, where an American went all around the world with the same ailment, got examined and suggestions for procedure to treat it. A VERY telling outcome was that in the US only the most expensive and complicated procedure was suggested to the patient. In other countries he was informed about cheaper and far less invasive procedures. Which btw worked and made him happy, so he avoided a complicated and potentially risky surgical procedure.
This is a problem with the US system. There are something like 2-3x as many surgical procedures that needed often simply worsening the well being of the patient. This is the flawed American thinking that good health care is the same as getting lots of complicated medical procedures.
The drug consumption is also astonishing. When using American health care I was shocked by how aggressively they pushed medication on me. People are trying to push medication on people everywhere in the US. Doctors, commercials, drug stores etc. You are not a patient. You are just a customer.
The US system really just leads to the health care sector focusing on pushing as many expensive drugs and procedures on the people who can pay for them an ignoring the rest, regardless of whether that brings better health or not.