That's also true. Different people have differing opinions about what should be national policy vs. state policy, and it can actually be counter-intuitive/paradoxical until you really look into it.
For example, most American conservatives favor strong national defense, but minority pockets of conservative areas are strongly anti-federal and favor local militias. It seems odd, but you'll find that most of that bent veer that way from both the Constitutional right to bear arms in a well regulated militia (unspecified as to who regulates it, but that's a small technical quibble), combined with an American leeriness about government (that the same amendment seems to voice) and you can kind of see how that kind of thing arises.
And there's similar sentiments all over the political spectrum.
As an American, the E.U. looks similar in this respect. Politics are tied up in originator theories (the E.U. was meant for xxx) and a European nation vs. local nationalism. With Europe being perceived as being very liberal from an American perspective, it's interesting to see politics that would be framed here as extreme conservatism as part of the normal modern European political debate and it's a good lesson that not everybody is wholly liberal or conservative on all issues.
As to what I support, this current overly complicated system, of overlapping but non-equal areas of trade, borders and monetary systems (the EEA, EU, ESDP, EFTA, EuroZone, Schengen, and on and on and on) seems unsustainable in the long-term.
It simply makes more sense to say "if you're in Europe, you are part of the EU which implies you are part of common defense, common trade, common monetary policy and so and so forth" and not worry about which part of the venn diagram a particular nation is. My prediction is that this will happen within the next 50 years. And 50 years after than everybody will look back and wonder what all the fuss was about.
From an outsider's perspective, Europe today is better than Europe was even just a couple decades ago and I think it's largely due to this unification movement. Having a larger government override local concerns seems particularly difficult for Europeans right now, but as an American we're quite used to it and find it generally tolerable.
As a frequent visitor to Europe over the last few decades, I enjoy the more unified Europe more, it's just easier to get around and do things, even if some of the local flavor has been lost (I used to love collecting all the various currencies, now it's just Euros, bleh).