In the grand abstract terms of the enlightenment, the legitimacy of government derives from the consent of the governed, and therefore no government should have the right to hand over its authority to some external body which is not democratically accountable to its own people. So when the framers of the EU arranged for the nations of Europe to do exactly that, they were repudiating the two centuries old political struggle for the rights and liberties of ordinary citizens, of government "of the people, by the people and for the people". It has always been my view that this was a quite conscious decision by the EU founders who, in the wake of two world wars, came to believe that the infamous national crimes of the 20th century could be traced directly to the democratic revolutions of the 18th century, and that the only long-term solution to this was to replace democracy with oligarchy.
However, I think there's an error there. For I'm not quite convinced that the democratic values of the enlightenment really ever quite took hold in certain continental states in the same way that they did here (and in the US which she discusses). Just as an example, France has always been run by an oligarchic bureaucracy, one much more closed to outside entrants than anything in this country.
So I'm not entirely certain that the EU is based so much on rejecting that enlightenment, rather, on those places which never really embraced it imposing their system upon those that did.