You can think of the UK system as "public money, public provision."
Again with some private supply on the side.
The US system is "lots of public money, lots of private money, mostly
Many other systems are "public money, private provision." In all cases
there are various complexities piled on top.
Singapore now is making some changes, outlined in brief here.
For the most part, Singapore is adding on some public money, but in targeted
fashion (one of the changes is for people over 90 years old, another is for
people over 60).
Here's from The Straits Times (gated, I write from the paper copy)
The first [priority] is to keep government subsidies targeted at those who
most need them, rather than commit to benefits for all. Universal
benefits are "wasteful and inequitable", and hard to take away once given, he
[the Finance Minister] said.
That's exactly the liberaltarian line and sometimes
the conservative line as well. It is a principle I strongly agree with.