Taxes as various European states and US states are sometimes on par. Everyone pays for somebody’s health problems, Americans as well, through insurance, it is just health insurance is mandatory in Europe. The other stuff boils down to effective use of tax money, it is easier to do it in a smaller state compared to US or Canada or similar. Individualism has an effect but at this day and age it is about lobby groups politicising any topic they do not like. FYI nobody likes to pay taxes.
Americans pay first through taxes, and then again through insurance, so it's even worse. Medicare+Medicaid cost about as much per capita as the UK's NHS, in part because they have intentionally been barred from being as efficient as possible, with e.g. limitations on using their negotiating power.
As for ease of doing it: At least several European systems does delivery via private actors, at least one has decentralised the insurance (Germany), several has segmented the public delivery in regional or local trusts or similar (UK, Norway). In other words: Universal coverage doesn't mean a single top down healthcare system, not is that necessarily desirable. E.g. the UK model uses trusts that prevents failure of leadership in one organisation from causing the whole to fail, and let's trusts get put under alternative management if they underperform.
If anything, the EU is a demonstration of how it is possible to do in a heterogeneous way across a much larger population than the US.
the NHS is not a shining model to emulate
It's underfunded, but it remains one of the top healthcare systems despite that, and provides universal care.
It's however relevant here to illustrate how insane the US healthcare situation is, in that Americans pay enough in taxes towards healthcare to provide universal healthcare if structured better, but chooses not to.
As an example of how to provide cost effective universal care, the NHS is a shining beacon. If the UK spent comparable amounts per capita to comparably wealthy countries to upgrade it, it'd be 20-30% more expensive, and a lot better, and still vastly cheaper than US healthcare overall.