I would love a Medicare for all, because that's the most cost-effective system we can possibly get short of having no system at all. It'd also increase job mobility--nobody tied to their jobs for "benefits"--and put a sane check the price of basic-level care.
However not all countries with public healthcare are drug-free--Portugal just manages, not solves, its problem despite 50 years of public health--and that's because the incentives to both sell and use these powerful painkillers don't merely go away with a public system. Medical doctors were also complicit; or the more responsible ones, yanked-away the prescription out of caution but unintentionally drove their patients to the black market.
I'll concede that a streamlined system would have helped alert the powers that be sooner, but I'm not convinced that the powers-that-be would have done anything concrete until it became a crisis.