+ The EU Parliament has very little influence on law making in the EU.
They do not create, write or formulate law - they have only the ability to reject it. Yes - they can work with the Executive to evolve legislation, however in practice, this is not really the case with most legislation. I'd argue that individual nations, via the Council etc. have more influence than Parliament.
Once again: the EU Executive is totally unelected, and fundamentally unnacountable.
2) The EU - post Treaty of Lisbon - is not a 'remarkable thing' - it's a failing system. From 1950-2000 it was more or less an economic Union, less so a political one.
Surely, European nations coming together post WW2 is 'remarkable' - but much of that is simply the 'new world order' , and of course the EEC.
Most European nations would be happy to participate in an EEC-like union of some kind - which was the basis of the 'remarkable era' that you describe.
Put another way: if Europe was right now, in a fully EEC-like Union, much like NAFTA, with visa-free travel, easy work visas, relatively frictionless borders, and some level of political cooperation - would Europeans vote to enact a political union? (Especially one in which the laws were made by an unelected group?) the answer would be a resounding 'No'.
The number of shape-shifting arguments that pro-EU folks have to get into to side step the democratic illegitimacy of the Executive is utterly astonishing.
Nobody seems to be able to provide a basic, decent argument for why in fact those who make the laws in Europe are not elected.
Most political leaders have to bend with the political wind, that's usually a good thing. Jean-Claude Junker et. al. do not care one bit what Europeans think, because they don't have to.