It always boggles my mind when people think moving around mainland Europe is a big deal. If you think of the different countries of Europe as equivalent to states here in the US (Alaska & Hawaii included), and realize that the US is more than TWO times the land mass of Europe.
Its like saying a kid has an opportunity to work in Boston, Chicago, NYC & DC here in the US.
Europe is a diverse region because of its different cultures. The US is not AS diverse, but we do have diversity of culture here as well.
Having the EU does allow for more ease of travel between countries, like we can between states. But since these are independent countries, they always don't have a desire to share the same culture or rules of a neighboring country. Independence & autonomy is a good thing!