In quite a few European countries the government isn't directly elected - so "unelected" is a bit tricky here unless you are extending that to those national governments, too.
Thing is that during the last 10 or 15 years (things seem to have accelerated after the latest big financial crisis) almost all consequential decisions taken at the national level have just been continuations of directives coming from Brussels, so even the politicians that happen to get elected at the national level are now just, I don't know what the best term would be, pawns? And this latest Next Generation EU madness has just made things worse, across the board, on account of "you'll be doing what you're told or no money for you!".
Looking at Brexit from that perspective helps make better sense now, it was not all a "the poor toothless people have taken Britain out of the wonderful EU!" as the tertiary sector elites in South-East England made it out to look just after the fact, as Britain was one of the few political entities that could have disentangled from the whole mess while they still got the chance for that. Not sure how to judge of the results of it all, maybe it's too soon to call it, but there was reason behind the apparent madness.
So that's (almost?) every government? For example, in the Netherlands, we do elect the members of parliament, but they then choose the government (hence the government being unelected).
And of course, that government then nominates a Commissioner, forming the European Commission together with other EU governments (hence it being similarly unelected).
And then of course, our government also gets a say about EU laws, together with other governments.
Just like the European Parliament, which does happen to be directly elected.
Also, a lot of important decisions are still national (even if they might be dressed up as European).
You mean like department for transport, or, you know, the national bank?
Sets interest rate, prints money, entirely unelected, the most important institution in the country.
Since you're referring to Brexit, you mean a political entity like the UK's House of Lords? A lot of the people in the pro-Brexit camp were often blaming EU unelected bureaucrats but seemed to be perfectly fine with their own unelected, appointed for life (and in some cases hereditary) lords.
I would like to understand why. I agree with you it's not all "the Russians' fault", and I may have jumped the gun a bit with my earlier comment. There's definitely a weakness in the disconnect between Brussels and the states/citizens.
Is it a fundamental belief that local government is "better" or more adapted to local circumstances at least? ("sovereignty trumps economy"). Or that a vote for the EU parliament is too "diluted"? Maybe, but being a part of the EU is a trade-off: there's a lot of advantages in being a part of a bigger trading and political bloc.
After the (IMHO) unmitigated disaster that was Brexit, very few other member states still have an appetite for leaving the EU. But it'd be better if the EU was something they'd enthusiastically want to be a part of, instead of begrudgingly. However, eurosceptics seem to have difficulty articulating the changes they'd want to see.