For me the "unelected" part is the key point here. I mean, if we ignore (totally valid) nihilistic views on the nature of governmental bodies, we are supposed to think that this is the point of modern civilized society (as opposed to "totalitarian states" like NK) where we rule over ourselves. We are even accustomed to call it "democracy" even though this is pretty laughable perversion of the word. But whatever, there is valid reasoning why "representative democracy" is the only solution, and there is valid reasoning why we need experts in the government, so long story short unelected institutions are ok and we are supposed to pretend that each of us kinda proxy-elected them.
(I mean, I'm really going to pretend it is ok: after all, I don't consider USA presidential election to be any more real than "proxy-elections" of appointed secretaries/ministers, and in Russia, where people actually vote for a president and he has a lot of power so it actually matters who is the president, people will claim for various valid reasons that this isn't really a democracy and Putin is more like a king now.)
But being as charitable to that world-view as I can be, EU structure feels way beyond that. Recently I tired to answer myself, how it came to be that I somehow "elected" Ursula von der Leyen? Or rather, how did "we" do it (keeping in mind that "we" as in "majority of voters" usually have different opinion than I do)? I just don't feel any connection. EC President is nominated by Counsel from the set of whoever-they-want (chosen behind the closed doors), and then voted-in by the parliament. AFAIK, the parliament never actually rejected a nominee, and BTW in Ursula's case 327/710 people voted against her, so she barely passed the bar. It somehow coincided that at the time my country's representative on the Counsel was "not really elected" as well, and by the end of my "investigation" I felt totally alienated. There is this European Commission thing, it affects me, I don't affect it. This is just how the world works.