The figure 10-15% of GDP of a rich country seems to be concealing something important: in the US, it's nearly 20%. In Western Europe, it's generally around 10%, for the same, slightly better, or in some cases much better health outcomes.

The UK has a famously stingy system which costs 7.5% of GDP, however the UK has better infant mortality than the state with the best infant mortality, Massachusetts. The worst states have infant mortality 2-3 times that of the UK. (Note that the poorest states in the US, LA and AL, still have GDP higher than the UK. So the 7.5% is of a much lower per capita GDP.)

The average EU country spends a little more than the UK, as a % of GDP. But has health outcomes well ahead of the UK, and considerably far ahead of the best-performing US states.

It seems kind of distorted not to point out that around half of the US spend goes either on luxury healthcare for the rich, or emergency healthcare to those who can't afford proper healthcare, or simple rent-seeking by entrenched players (pharma, doctors, insurance companies) and that none of these improves health outcomes by very much.