You are right for pointing that out (about him being born Vito Andolini and then being renamed Corleone by the clerk, even though the interpreter told him the boy’s full name).
That was back when the United States and Canada were both regarded as places for immigrants where you built yourself a new life and future, as opposed to nowadays with people wanting a free lunch (totally agreed also with your point on today’s system making chaos).
"My father's name was Antonio Andoliniand this is for you!"