Each member state elected their own government, which in turned determined their head of state by their own country's rules, then those heads of governments appointed VDL.

Do we even have a real choice with our governments? As a minor yet dramatic exemple, last year in France the Front Populaire won the last parliamentary elections, and according to republican tradition and common interpretation of the constitution, should have formed a government. But that was refused by Macron who named yet again an even more impopular proto-fascist government because it's technically in his power.