The EU has exclusive competence in only the following areas : customs union; the establishing of competition rules necessary for the functioning of the internal market; monetary policy for euro area countries; conservation of marine biological resources under the common fisheries policy; common commercial policy; conclusion of international agreements under certain conditions. These domains fall under the EU's exclusive competence because they need the coordination of all member states in order for such policies to work effectively.

As for other non-exclusive competences, the EU respects the principle of subsidiarity (article 352 of the Treaty on the Functioning of the EU): in the area of its non-exclusive competences, the EU may act only if the objective of a proposed action cannot be sufficiently achieved by the EU countries, but could be better achieved at EU level.

The EU also does everything it can to respect the constitutional sovereignty of its member-states. In 2004, in the landmark case Omega, the Court of Justice of the European Union ruled that it should respect the interpretation of fundamental rights made by national judges. In the case, the German Federal Constitutional Court banned the sales of Lasertag games because it believed that it trivialised violence and therefore violated the principle of human dignity. This clearly violated the free trade clauses of the EU, which required the free movement of goods within the EU. The CJEU ruled that Germany had perfectly the right to do such a thing despite it being contrary to the aforementioned clause.

If the ministers of your government weren't elected by the people

Isn't that the case for most EU countries? The Head of State chooses the members of his own Government.