I don't know where you are from but in my country I never voted for the President of Republic. I can't: the President is elected by the Parliament, which I vote for. I also never voted for the Prime Minister. I can't: it's appointed by the President. Despite currently being the leader of the largest party, her closest predecessors have not been the leaders of the largest party since 10 years ago. Actually the President can appoint anybody, there are not even rules about citizenship, place of residence or being elected. The previous three Prime Ministers were not elected by anyone. By the way, a government is still government even if it does not have a majority support in Parliament. It will have a hard life doing anything but technically that's not a problem.

In the EU we have the Parliament, which is elected by the people, the Council which is composed by Prime Ministers, Presidents, whoever rules each country and the Commission, which is proposed by the Council and elected by the Parliament. Not so different from what I see in my country.