I think we can look at the previous issues with Obama to get an idea of the effect of Bernie's policies. e.g. remember "Death Panels"? That's categorically worse than the Bernie doctor thing. "Death Panels" was literally believed by
about 40% of the USA's adult population in polls only 6 weeks before Obama's re-election in 2012. And those polls had held steady since 2009, so the number seems reliable. So, uh, since Obama survived almost half the eligible voters holding a steadfast belief that he was literally instigating forced euthanasia for the sick and elderly, I think they'd have to go pretty damn far to smear Sanders on healthcare.
Also, on the Sanders thing with how we pay for Family Leave: Sanders is supporting the existing legislation which would create a social insurance scheme similar to Social Security to pay for it. That is: every pays in, and everyone is eligible for support if they need it. So the message is "you worked for this, it's your entitlement". Whereas Clinton is keen to push her populist line about not raising taxes 1 cent on the middle class, so she is attacking the existing legislation that is being considered, and wants to make here own one, where it's paid out of general revenues and they raise taxes on just the wealthy to pay for it. But payments made out of general revenue are much more easy for a future government to just snatch away, especially when they can argue "someone else paid for this, you didn't earn it". Sanders is supporting FDR-style legislation and Clinton is supporting populist-style legislation. Her version is a vote-grab and it's basically flawed because it codes it as "someone elses money" rather than a social insurance entitlement that nobody can take away, because YOU paid into the scheme.
EDIT: and here's a theory: just perhaps the sheer intensity of the "Obama = socialist" rhetoric itself has made it possible for an actual socialist to consider running. By not holding back whatsover on the anti-Obama rhetoric, the Republicans might have actually blurred the lines between moderate progressivism and actual socialism enough that it's now viable to run as one.