Well, that's poetic, but doesn't mean it works out that way in the real world.
For one, America was not actually founded solely on "tired and poor masses yearning to breathe free", but on a huge takeover and displacement plan for the native americans.
Then, "tired and poor" masses of african americans were used as slaves, and tons of "tired and poor" masses of white people were also used quite like slaves (sharecroppers and the like).
And even after that stopped, there was tons of racial tension (do you think Italians, Irish and Jews were welcomed to New York initially?).
Another consideration, if tens of millions of muslims, for example, come to the US and change it vote by vote, to follow islamic law, would you like it? Or does you "bring me your..." proposition presupposes that mass immigration will not happen if all is left open?