'Public' bureaucracy is often just as efficient as 'private' business.
If not more so, since private business puts its own profits far ahead of the welfare of the patient.
The reverse of that is politicians often put their own personal interests (political profit, etc.) ahead of the welfare of their constituency (the patient in this case), which makes the public system more inefficient. I'm not saying there's no validity to your point, but that those who would be running a public system wouldn't always be philanthropists.