The EU budget is just 1% of GDP, right? If people saw they pay a few hundred in taxes to the EU, and several thousand to their national government, they would stop complaining that the EU is too expensive. "It's the statue, man, The Statue."
So your suggestion would be met by slobbering cries of 'Taxes bad! Bad! Bad!' from people whose idea of being balanced is an ability to drool on an even keel. (And I don't just meant the ones who write for the FT and the Economist.)
In fact I'd add the influence of Anglo-Saxon-ism to Nanne's list, because ideologically it's the only thing that has the potential to tear apart the EU.
The infection is already running deep. Wwhile I'm sure there's an immune system of sorts in place, I'm not so sure it's strong enough to nurse the patient back to rude health without the need for a more serious intervention.