WW2 did put US as the global economy giant because it was the only country without a destroyed infrastructure. US supplied the money and goods to help rebuild the west and certain parts of Asia. Naturally the standard of living for the US citizens, as well as the US economy skyrocketed as a result.
As far as US dollar being in constant demand, that has nothing with WW2 and everything with the US making a deal with the saudis to trade the oil only in USD in the 70s. Since then every oil producing country that wanted to sell in other currencies was invaded, thus securing the constant supply of USD demand.
Europe is fully on board with this deal, they even help with military operations as coalition forces. So as much as US benefits from this deal, so does the EU.