You're describing capitalism and the global movement of labour that long predates EU freedom of movement.
But, EU freedom of movement exacerbates the issue (the existence of a brain drain within the EU is not a particularly controversial viewpoint and has been widely covered in the literature). Especially when tied to economic and mercantilist policies emanating from the core countries that beggar peripheral countries and deliberately contract their economies, using debt as a form of control.
It is just another form of exploitation, sadly. The EU should be an organisation that sets out to alleviate this problem. Sadly, it does the opposite.
Your charge is that the EU is exacerbating the diminution of the economies of the 'peripheral' states.
So... genuine non-rhetorical question... Looking at Greece's economy now, how do you think it would compare if Greece had not joined the EU in 1981? My impression is that, in relative terms, Greece's economy is stronger now than it was back then... and similar for other 'peripheral' countries.