What would be the incentive of majority of people flocking to EU services ?
Not really anything unknown about this situation. There are many obvious historical incidents.
The way the US bill of rights applies to non-citizens not residing in the US is complex, but considering how little the US courts care about the question, functionally we have no rights when dealing with the US government.
This is the lynchpin to things like seizing people passing through US airports and holding them indefinitely in Guantanamo bay.
At least my own government can't get away with treating me like a brick.