The cost is the cost. If the hospital, doctor, etc, reduce or eat the cost of treating an uninsured patient, they just pass it on to those who are insured. If a big company gets discounts, similarly, the costs just get pushed onto other companies. Our health care system is a "for profit" operation. As long as there are shareholders, the health care business must reflect their interests which may or may not reflect those of the patient.
While I waited in the emergency room to be treated for my dog bite, I determined (based on observation only), that I was the only one their for due to a true "emergency". The others appeared to be "sick" but not in need of emergency care. If you have no insurance, you go to the emergency room for treatment. I don't believe anyone is turned away. All are treated regardless of whether they can pay or not. The cost of those who can't is just borne by those who can. Sounds like socialized medicine. The problem is that it's a very inefficient system. Emergency rooms don't do preventive care.
Hospitals, doctors, labs, etc. don't "eat" the cost for treating the under or un-insured, they just pass the cost on. If medicare gets a discount, the unrecovered costs are passed on to private insurers and ultimately those of us paying premiums. A single payer system is the lesser of available evils.