Yanis Varoufakis describes the structure of the EU as "We the governments" rather than "We the people".

Indeed, the EU is currently a union of countries - that is the way it was designed: the founders were nationally attached. I would love to make it an actual country, but I am in a minority for now - most citizens of Europe still feel they belong to their nation before belonging to the union... I dream of that changing: when one has traveled a bit inside and outside of Europe, one realizes that national identities are irrelevant compared to what European nations have in common. But that would be a radical redesign of the EU.

I'm sort of where you are in that I don't really feel like I'm a part of any nation, I'm just a human. I do feel Terran however (ie like I belong to Earth and vice-versa). So I feel European as much as British.

As a subject of Her Brittanic Majesty however, I'm about to have that citizenship forcibly removed. The only other option appears to be to become a political refugee. Like if a third of people in your region voted and suddenly you were no longer to be a national of the country you were born in; it's weird.

I can't help wonder if this is an age and class thing.

Is one is younger and hold higher education, one can more freely move between nations.

But if one is older, more settled with family and such, and perhaps has done industrial work for most of ones life, i suspect one see less benefits from EU. Heck, some of the directives may be seen as disrupting hard won job security.

Seems like a reasonable hypothesis.

Yes; in political science terms the EU is a very interesting hybrid between an intergovernmental organisation ("we the governments") and a supranational government (which could theoretically be "we the people").

The structure is very close to that of a very powerful intergovernmental body, though, and there's not (yet?) a strong European polis/sense of shared identity. The democratic aspects of the EU as itself (rather than as a confederation of themselves-democratic national governments) are underdeveloped.

"sense of shared identity" may be a bit loose; I meant in the sense of not feeling a constituent part of a common political community. Plenty consider themselves European (I suspect rather more in the UK since the Brexit vote made the question a bit more salient) but I doubt there are many whose principal allegiance is to the European Union, rather than France, the UK, England or Yorkshire. And yes, this is an informed assumption rather than based on a particular survey.

In practical terms, as someone else stated, European Parliament elections are conducted as a series of more-or-less isolated national campaigns, with a weak sense of a body of citizens making a decision as a continental group.

This is perhaps beginning to change, with the Europarties nominating explicit lead candidates for the presidency of the Commission at the last Parliament election, which provides a certain cross-border unifying factor. It's notable that in my country (the UK) these candidates were barely mentioned, though, and the race was treated as a proxy fight for future control of the Westminster parliament, or as a chance to elect eurosceptics to protest the EU as a whole.

Sure but don't conflate the EU with Europe, much as the former would like you to.

A lot, but a minority.

And it seems to depend on where in a nation one lives, and what occupation one holds (or held).

He is not wrong. Most EU institutions represent the member states rather than the European people directly. The Europarliament is the one exception, the only institution directly elected by the European people. Still limited by national borders, though; each country gets a number of seats according to their population (rather than the number of votes), and you can only vote for parties from your own country.

Still, it's easily the most democratic part of the EU government, and should definitely have more power.

If they got it, would they not become the subjects of more lobbying and therefore become less democratic?

It is easy to represent someone when you don't have much power. It becomes harder when you have power. Maybe different people seek to become MEPs for example. As today it is not a very highly regarded political job IMHO.

Perhaps, but they are already subject to a lot of lobbying. So much so that it has happened that two different parties propose the exact same law that was clearly written by lobbyists.

But with greater power comes also greater media attention, I hope, and therefore more accountability.

I think he was in a unique position as an "outsider" desperately trying to do his best to help his country and indeed Europe as a whole.

I've read a couple of his books and I've definitely become more skeptical of how the EU is currently structured (although still a Remainer) and much more sympathetic to the plight of the Greek people.

Do you care to elaborate? Given the power imbalances especially regarding the European Parliament, his statement is not wrong.

Moreover, given his political work, we can safely assume that he gained quite a lot of insights into the inner working of the EU. So his personal opinion that results from the experience is anything but irrelevant.

Kinda. He has some deep insight into (macro-)economics, but i fear he is too enamored with the EU project as an intellectual.

There's a belief that legislators elected as MEPs are not particularly high quality, so it's not a good idea to give them too much power. OTOH it's hard to convince ambitious talented people to run for a job with little power, so there's a catch-22.

It's also a way around the democratic deficit, though in reality the deficit is in government to government horse trading with plausible deniability ("the EU made us do it!"), not anonymous bureaucracy.

Thanks. It is a pity I am not aware enough of how European lawmaking works.