It's an interesting model. It was more like the EU in it's inception than today in that each state was it's own country under a thin umbrella of federal government. The outcome of the US Civil War changed that significantly, but not enough to have a coherent, omnipotent federal system. (The EU doesn't have one either; each EU country controls most of their own sovereignty). One main difference is US states can't secede from the US while EU countries can.
Another issue is free borders between states. If state A enacts universal healthcare, how does it prevent the sick inhabitants of neighboring states B,C,D,E from emmigrating and bankrupting the state? This model poses all sorts of problems. Even colleges have to deal with this. If you live in the same state as the college for a year, you are considered a permanent resident and get a significant tuition deduction (like 75% less). There are hundreds of examples of where autonomous states with free borders cause administration headaches.
Imagine if the EU decided to have a governing body that took taxes and distributed services, like healthcare, military, infrastructure, etc. It would have to cut out parts of the tax code for Greece and add parts for France, etc. Not only that, but by industry. Then tack on special interest lobbying. It would be a mess. Not only that, what if 1/4 of the countries didn't have universal healthcare?
Most of the reason why Americans get so frothy about tax is because they've made it ridiculously complex to comply with.
For another example, take getting a driver's license. In the UK you fill in a form online, they extract any other relevant details from your passport records and put in your credit card details and they send you a license within the week. It's a 15 minute process. In America I got to finding out I had to go to the DMV (and that I should go to the DMV in another city because it's somehow superior) and promptly gave up.
I know you said aside from healthcare, but even if you had the hypothetical best insurance available, with no co-pay or deductible, just keeping track of all of the arcane and opaque numbers, policies and restrictions is a huge administrative burden.
Now to be clear, I understand why some of these systems are the way they are and, in some cases, can see the necessity, but the fact remains that in almost any other developed nation there simply wouldn't be an equivalent administrative burden to just existing.