If you want a counterexample to the idea that healthcare prices are high solely because the demand for healthcare is inelastic, look at dental care. There isn't a lot of government funding for dental care, although it is very much necessary for your overall health. As a result, dental care is comparatively cheap - a full routine cleaning plus a check-up from your dentist (handled by private insurance) costs about the same as a single lab test (handled by medicare/medicaid and private insurance). Serious interventions, like wisdom tooth removal, crowns, or braces, cost only thousands of dollars. By contrast, routine surgeries are often in the tens of thousands.
Similarly, in education, private trade schools tend to cost $10,000/year or so, while private colleges are $50,000-70,000. The economic benefits of a private trade school are actually similar to the benefits of several majors.