The argument that the EU has brought peace to the continent is frankly laughable. If anyone brought peace it was Oppenheimer as the threat of his little invention meant MAD.
While I generally agree with your comment especially the parts about the EU being undemocratic, I think no one can state with any certainty what caused something _not_ to happen.
Answering only "MAD" assumes that Germany would have ceased to be a threat to European stability, although it had been frequently at war with its neighbors for centuries and had committed crimes so utterly vile that people will talk about them a thousand years from now.
But let me first point out that within Europe, but outside of the EU there has in spite of "MAD" been wars in Europe:
The break-up of Yugoslavia (not just deemed a civil war, since it did involve NATO) and now also the Russian invasion of Ukraine. Putin has promised that if necessary, Russia will come to the rescue of the significant Russian minorities in the Baltic countries and while it is most likely just his usual saber-rattling, it is being taken quite seriously and is something that is unlikely to trigger the Mutually Assured Destruction, thus a scenario that the MAD cannot rule out.
Secondly, and as first stated, one cannot just write off the concern that Europe would be unstable without the integration of Germany. A major cause for WW2 was exactly because after WW1, Germany was just told to make reparations and to not re-arm itself. Luckily, a smart person in the US, George Marshall, was able to convince everybody else not to repeat this mistake and instead implement his plan. The importance of the Marshall-plan for European stability cannot be underestimated.
This platform of stability allowed for the creation of and has been taken a step further by the EU. Now where millions of EU-citizens have moved within the union (creating lots of three-passport families) a war among EU-members is unimaginable. This is something new in European history and I think one should not underestimate the significant role that the EU has played in this.
Btw, now that you specifically mention Oppenheimer, I will mention that I actually worked at Los Alamos National Laboratory for a couple of years, in what can fairly be described as a spin-off of the Manhattan project. With Edward Teller as one notable exception, I don't think the intimidatingly smart people behind the fissile and fusion bombs can be said to have been in favor of MAD.
To maybe give this post a shred of relevance to the actual topic, I will add that European integration is no simple matter.