There are some distortions told about universal coverage and single payer
Firstly, these things are different things.
You can get universal health coverage without single payer. For example, Germany, Switzerland, Belgium get universal health care/coverage through insurance mandates.
Secondly, a lot of countries that have universal health coverage have dual systems.
----Government sponsors some amount of catastrophic healthcare and individuals can buy additional healthcare. (Singapore)
---- Single payer covers some part of healthcare, individual private health insurance covers the rest (France)
There are some countries that do get to universal health care through single payer, and these are Canada, UK, Scandinavian countries. However even with single payer rules vary by provinces in Canada and the UK.
We can argue about the best system, but arguing that single payer is the only way to get universal coverage or the only way other western countries do it is just wrong. Some western countries do single payer, other do dual system, and others do insurance mandates.
Sources
https://www.vox.com/cards/single-payer/
https://www.forbes.com/sites/danmunro/2013/12/08/universal-coverage-is-not-single-payer-healthcare/#680a046e36ee
https://truecostblog.com/2009/08/09/countries-with-universal-healthcare-by-date/