Which of the following approaches for providing health care in the United States would you prefer: replacing the current health care system with a new government run health care system, or maintaining the current system based mostly on private health insurance?
You see a lot of ill-designed polling questions, but this one actually manages to exclude the major alternative to the status quo, namely a system similar to Medicare, or the health systems of many foreign countries, where the government doesn’t run the health care system but the government does run a health insurance plan in which everyone is enrolled. The distinction is semantically subtle but absolutely crucial. In the United States, state and local governments actually run school systems much as the federal government runs the Post Office. In England, similarly, the government runs a National Health Service employing doctors, nurses, etc. running hospitals and other clinics throughout the nation as a government agency.
A very different alternative, however, is to simply have the government run an insurance program that will pay (in full or in part) for (some) medical procedures and services, while still leaving health care providers as private for-profit or non-profit institutions. This is, overwhelmingly, what counts as the “left” position on health care in the United States — government run insurance not government run health care.