Just back in 2010, Finland and Iceland were the two Nordic countries with the lowest total tax revenues as % of GDP, and the tax level in a country goes up and down depending on whether the country has a conservative administration or a social democratic administration.
No it doesn't. Increase in tax revenue relative to gdp since 2010 is more likely due to post financial crisis recession and declining gdp in those countries. Finland has had a conservative administration since 2010, which is probably going to change after elections in April.
Exactly as you state, 5 percentage point difference is really significant between countries if both still cover most of the healthcare with tax money.
You have significantly lower taxes and fantastic public heathcare (other services too). But the reason for that is the economic prosperity itself, not the model of your healthcare system. Healthcare expediture per capita in Finland is 3013 EUR per annum, Norway 4653 (OECD Heath at a glance 2018). That is a significant difference, although large part of that money is paid as higher salaries to employees. Healthcare expenditure is between 9-10% of gdp in both countries, but Norway can do this with a lower tax rate. Finlands system is actually not a single payer system, but OECD statistics take this into account and are reliable afaik. Finland is by no means poor, but the service level is significantly lower despite higher taxes.