Playing devil's advocate, I guess one could speculatively argue that Ireland's private system* brings up the overall average while the public system only caters to the poor and deprived, but I don't think this would be enough to bring the averages up on its own. I'm sure it's more likely lifestyle factors.
* for those unfamiliar, Ireland has a two-tier health system with extremely poor, underresouced single-payer free healthcare coupled with privately-run hospitals for those with the means to pay.
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.
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.
Why? Ireland has longer lifespans and better health-adjusted life expectancies than the UK.