There are radical left and radical right movements
But EU has to exist from one cardinal reason
There was not war in EU area all that time when EU exist
Memories about WW2 are still alive and people in Europe don't want that happen again
70 years of peace in the EU.
This between countries that were at war for centuries and even before these same countries were even born the people living in those territories were at war. A bloody history is nearly coming to an end and people still think is bad.
The EU is a nightmare for every other great power.
Russia is scared as shit of such a Union, in their entire history, they haven't been able to win a war against a coalition from Europe without help from the same space, they can't even to dare to think of a war, be it traditional or economical with a union 4 times as big in population and 10x bigger in economy terms.
The US is also in the same position, China thinks the same, a fractured Europe is the best thing they can dream of.
Yeah, it's the least integrated but not because of those.
Imagine driving from NY to Washington and in Wilmington, you have to change lines and drive on the other side of the road.