Well, I'm also a full-blown federalist, although I understand very well why people don't like the idea. I'm still hoping to be able to hold a genuine European passport in my hand that states "European" as nationality and "German" as mother tongue. Here is the main reason why I favour a genuine union with a partially joint military, a strengthened power for the European Parliament, and less power for appointed commissioners in the future:
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.