Yeah, this is what I would've expected (hence the devil's advocate qualifier), which is why I think our life expectancy has other causal factors.
Although, one thing to note is the the government does invest in private hospitals as well, so that 79% figure won't be exclusively public healthcare. This is due to public-private-partnerships whereby a single site will accommodate both a public and private hospital with the entire project being funded by a combination of public and private funds, usually with an extremely favourable arrangement for the private side.
Sounds like the public system provides the majority of care.
Any "public" system still has a private-tier available to anyone that can afford a plane ticket.