Europeans tend to credit the EU/EEC for the peace, but as an American, I find that totally implausible. The peace was because Europe was divided into two vassal regions and the actual superpowers decided not to go to war because of MAD. Now that the Cold War is over, we've already had a series of wars in the Balkans and various wars in the Russian periphery. True, France and Germany have taken a break from fighting each other for a long-ish stretch, but I think that trend would continue even with a Frexit because it's mostly built on memories of how bad the last two wars were.
MAD and the cold war prevented war between countries on either side of the iron curtain, sure.
But then you go on to claim that war among countries on the Western side was prevented by memories of war and not the EEC, without any reasoning as to why. I don't buy it. The first World War was already terrible, yet these countries were at each others' throats only a few decades later.
Russia and Ukraine are not EU/ECC members, and neither were the balkan states back when they balkanized. Wars outside the EU don't disprove that the EU plays a major role in bringing pace among its members. I would agree that it didn't necessarily bring peace to all of Europe, but that's a stronger statement than most people intend to make
It's impossible to prove causality, but Europe has never seen longer and more widespread peace than the last 70 years.