In virtually all empires and great geopolitical entities there is always a "home nation" or region associated with it. That region holds profound influence on the overall culture and actions of the wider entity. In the USSR, that home region was Western Russia. In the UK that entity is South-East England, so called the "home-counties" which is very telling (Scotland, Wales, Northern Ireland etc are obviously not home). In the Roman Empire that home region was Rome. In the USA that home region is the East Coast. The list goes on.
The "home region" of the EU is pretty much Belgium. It works well as a home region because Belgium is very strange. There's almost no "Belgium" in truth, more like two nations stuck together like the UK; it just so happens it's a union of Romance and Germanic peoples which works splendidly as a kind of home region for the EU as it is presently i.e. before the Slavic peoples become more important. I hope Belgians here aren't too offended by me claiming Belgium doesn't exist (I'm sure it does, it obviously does in Brussels) but Belgium has always fascinated me for that reason.
Also, when I said "two nations stuck together like the UK" I do know the UK contains more than two nations, I'm just saying the relationship is similar.
Why Paris? Strasbourg is far more connected to the EU than the French capital ever was.
Strasbourg works well too. I do lean in favour of the independence of Alsace obviously so maybe Alsace would work well as another home nation of the EU. Well, "Western European Union".