There are a number of reasons I'd question this line of reasoning, but the biggest one is that many of us did not have a choice - our names were changed without our consent.
Oh yes, I am by no means certain about my line of reasoning, but emotional intelligence and self-awareness are inheritable traits; so even if your parents or grandparents changed their names, it would still indicate an increased likelihood of your possessing these traits.
The point is that a lot of the time immigrants who received Anglicized names did so not because they chose to, but because they happened to get an immigration officer who gave them an Anglicized spelling on Ellis island.