Up until 2005, roughly 10% of the population couldn't even access healthcare, at which point the PRC built out more care centers and invested in training more doctors, but there's still a significant shortage, such that scalpers sell outpatient appointment tickets for 10-15x markup over the actual appointment cost.
There's plenty of ways the two countries are different, but healthcare seems like an odd choice to try to "one-up" the US on, even if its programs like medicare, medicaid, social security disability and others still leave gaps.