People won't use such a scheme because it's too complicated. And the people doing medical research need everyone's DNA in a big database so they can analyze it in bulk. There's relatively little scientific benefit in just a few people's DNA. And that concentration of information is where your concern is.

How about, if you want socialized healthcare, you pay the cost of privacy. We guarantee to pay for your healthcare, but your DNA is freely available in everyone's database? Signing up for Medicare/Medicaid? Submit a DNA sample. That will give a huge population for study and the rest, people who aren't on socialized healthcare, can protect their DNA if they want. If the government is paying for your healthcare, you already lost your privacy to them anyways.