At one time, that phrase had a purpose behind it other than empathy for the foreign downtrodden. There was a substantive need for people from other countries that actually wanted to create a better life, expand our (America's) horizons, and to become American.
The opportunity, the wide open spaces and the hard work involved then pretty much forced self-independence of some measure. It also facilitated assimilation to American values, whether they intended so or not.
Today? In truth we don't even really need even the best of what the rest of this world's population has to offer, let alone its dregs. And, by dregs, I mean the third world sh!thole denizens for whom this country is so adamantly cultivating entry and citizenship.
Instead of H1-B visas, asylum/refugee green cards and the like, we should be instilling real education, drive for accomplishment and success for our own children so they can fulfill our future needs instead of some Muslim Terrorist or other foreigner just out to bleed our entitlement system dry.