The way people in the policy community see it, this is totally backwards. Almost everyone who has health insurance in the United States gets help from the government to afford it. For the elderly, that’s Medicare. For the disabled and the poor, that’s Medicaid. For full-time workers it’s the tax subsidy for employer-provided health insurance.
Some of what you see in this poll is a simple misunderstanding — older Americans either don’t know what Medicare is or mistakenly believe they have "paid for" their benefits with earlier taxes.
But Americans who get insurance from their jobs are also benefitting from a massive government program. A program whose existence is hidden from sight but is nonetheless quite real and substantial.