I'm talking about Universal Health Care systems (for clarity: totally free healthcare for residents in that country), not Public Health Insurance systems.

Europe is unfortunatelly also riddled with the latter system and having lived in countries with one kind and countries with the other, they're quite different and the system with Insurance is invariably worse in terms of denials of coverage as well as cost (also because nowadays they all have laws that force every resident to have health insurance, which as result is more costlier than before those laws - as I saw first hand when I lived in a country with such a system when such a law came into effect), whilst UHC tends to have longer waiting lists (think 1 or 2 years of wait for some cirurgical procedures).

Absolutelly, some of the absurdities of the US system are also present in the so-called "Mixed" Systems (i.e. the ones with healtcare insurance but more regulated and with a public option for some) and if you look at the kinds of governments in those countries for the last 3 decades, you'll notice they've been invariably neoliberal mainstream parties (setting up such systems is part of the broader tendency in Europe to privatise just about everything that has been going on since the 80s and was copied from the US).

IMHO, except for the long waiting times, the problems with Healthcare systems in part of Europe are the result of them having been transformed to become more like the US system in the last 3 decades.