I don't think the EU is a recipe for peace or is making Europe more peaceful. Far from it. The EU looks a lot like a bunch of very unhappy neighbours kept in line by the threat of wildly over the top retaliation from the federalist core if they even think about stepping out of line.
Look at Brexit. The EU is talking about grounding all flights, banning British firms, insisting on obedience, suggesting that the UK be split up to convenience them and demanding huge ransom payments (and don't argue with me about this, these are my perceptions). Then you look at eastern Europe, Catalonia and so on.
I do not perceive the EU as a happy peaceful entity. If it looks peaceful it's only in the sense that a successfully dominated population doesn't fight back. True peace comes from understanding, collaboration and compromise. The EU sucks at all three.
Also bear in mind that the vast majority of all British politicians were "Stay", because they knew what's at stake.
It is also mysterious to me what makes you think that the EU would "suck" at understanding, collaboration and compromise. These are central pillars of the EU, without those three factors the EU could not have possibly been founded by unanimous votes of all 28 member states. All I can say to people like who - who genuinely leave me baffled - is that you could inform yourself better about the EU structure and what EU programs and institutions are actually doing and, most importantly, how incredibly cheap the EU is in comparison to the respective national budgets.
Last but not least, there is either peace or no peace. People, states, and nations will always disagree about certain matters. Your idea of a "true peace" is naive and unrealistic.
We live in different times today. 2018 is not 1918. I would be worried if younger generations could imagine to waste their lives and their wealth for war.
The problem is that too many of the younger generations trust the older generations (current politicians) which trust in lies and wars as solutions. E.g. Syria, Libya, Iraq, Israel.
US, and to a lesser extent European countries, have had little difficulty finding bodies to send to war in the last 50 years.
That claim is hard to square with the literary canons of a number of European peoples, the creations that these people consider the bedrock of their culture and which are exported around the world as what that respective people has to contribute to humankind. You'll find that these canons are to a large extent dominated by nationalists, especially in the wake of 1848. Sándor Petöfi, Mihai Eminescu, Taras Shevchenko, and Elias Lönnrot were not international cosmopolitans, at least not in their literary activity.
The EU is in my opinion a great success, way greater than what most people realize. Previous generations, who still remember WW2 or its effect from their childhood, recognize what kind of monumental achievement the EU was - which is based on unanimous votes by member nations with varying governments from all political sides!
Unfortunately, younger generations have lost sight of this, some of them erroneously believe that wars could not start again in Europe without the EU. That's very short-sighted, historically, Europe has been torn by wars almost all the time.
Although the EU is certainly not a warrant for peace, it makes a peaceful future overall much more likely in the long run. Take the cordial relations between Germany and France as an example, without the EU these would not look nearly as good as they do now.
Besides that, there are also economic reasons for the EU, of course, but I just wanted to mention this perspective which is all too often forgotten today.
So what about different cultures? Does the EU "destroy" local culture? About that I have to unfortunately point out the sad truth that nationalists generally tend to lack culture. Few outliers aside, the people who actually advance culture have practically always been international cosmopolitans and humanists. And there is a reason why we all don't speak Latin any longer. So I don't buy the "culture" argument from nationalists, it tends to be very hypocritical.
Finally, as for the priorities you mention, sadly there is a case for technocracy to make here. The impacts of terrorism and immigration are minuscule and almost neglectible in comparison to the impact of, say, advances in technology, and the public's fear tends to be badly aligned with reality.