Single payer has many, many more data points from which to draw conclusions.

Well, it tries to. Whether or not it succeeds is the subject of lots and lots of debate, but single payer systems I'm aware of have accountability models driven by (usually public-sector) audits and quality controls that try to optimize for health outcomes. The rationale being that the payer (government) spends less overall, and is more likely to be supported (voted for), if the population has good health outcomes in areas that the government can influence via the healthcare system.

Paying and quality assessment don't need to be coupled. The government doesn't need to be paying for the healthcare in order to make recommendations aimed at improving health outcomes. It would probably be happy to swell its ranks by starting up a dental review board.

Though I'm not optimistic about either its ability to fund healthcare or assess its quality.