The goal of profit-seeking companies is profit maximization. And profit has to come from somewhere. For a private company to provide health services for the same cost as the public system, profit margins must come at the expense of facilities, or levels of patient care, or staff wages and working conditions.
Privatization simply means more public money going to profits instead of patient care.
And frequently, the drive for profits forces costs to go up - beyond costs in the public system. Lessons from around the world tell us it is not. Generally, privatization simply draws personnel and resources out of the public system into the more expensive private system. In short, privatization just shuffles the queue instead of getting rid of it. In many cases, it makes the queue longer.