Wiping out the native population is a very sad moment of US history that doesn't invalidate my point.
Who says immigrants with a different skin colour don't move to Europe because of the ideals? That is bollocks and borderline discriminates other cultures with the assumption that they do not care about becoming European. The incorrect assumption being that it's never been the people that convert to a culture, but instead the culture evolves over time to accommodate the change in demographics. Racial and cultural purity is an ideal that never really existed. It's easy to imagine the citizens of Rome complaining about the Celts and, yikes, the Barbarians ruining the pure Roman blood two thousand years ago.
The future of Europe, and the rest of the world, culturally and genetically, is a melting pot of people from all over the world. To me it is an absolute win, especially since history proved time and time again that the more people with different experiences and lives come together, the more society advances.
Disclaimer: I am definitely biased as a European-born mixed race citizen. The more you mix the cultures, the more arbitrary those borders and ideals seem.