Certainly not all EU Governments are elected. In The Netherlands, for example, you vote for Parliament. Parliamentary leaders then seek an agreement and propose a cabinet, which is formally instated by the King. When you cast your vote, you don't know who will be in government, nor do you have any say in it.

I'm a proponent of this system, just wanted to point out that governments are not necessarily elected. I think the same holds in many other EU countries, e.g., Belgium and Germany.