No. This idea is the result of decades of propaganda.

The government has to answer to the people. PE firms do not: they answer only to money.

Worse than PE? I very much doubt that but I guess the answer will depend on the voters, how the healthcare ventures themselves are structured, how much open corruption the local culture tolerates, and similar.

However AFAIK most government solutions in the west involve public "insurance" as opposed to the direct operation of health care facilities themselves.

Why?