"Brexit shows what we have in the EU. She clears up issues among neighbors, grating them into rules and sometimes unsatisfactory compromises. But she grates them small, in a civilized and peaceful way. That's her big achievement." (rough translation by me, sorry)
https://twitter.com/tagesschau/status/1105932622000209921 (in German)
It only became fully realized with the Maastricht Treaty in 1992, which was shortly after the fall of the USSR in 1991.
But I wouldn't go as far as claiming it's solely responsible for it because rarely anything is as simple as that.
I am not sure how true it was but I heard a theory that Europe lost much of its intellectual prominence due to so many young men dying in the trenches in WW1 - now Europe does good intellectual work now but it is more "neat thing out of Europe" as opposed to "the hotspot". I am not well equipped to evaluate the merits but the idea is interesting at very least.
https://www.politico.eu/article/eus-milk-scramble-for-africa...
I don't think it's related to the EU as institution
Do you get a vote over antitrust regulations enacted by your own government?
You usually vote a party which might (or not) have a high-level guidelines over a few select policies, but otherwise trust them to make the best decision based on how their political view matches with yours.
You elect EU parliament member the same way, and you are free to vote otherwise if you do not like their work (and yes, the actual effect of your individual vote is statistically irrelevant, just as the vote for your own government).
Besides, the point is moot, because parliament members tend to group themselves based on shared political views, rather than by country.
That countries with more population should have more representation and more weight in the decisions, to me makes sense.