This rhetoric is fraught with problems. It is not disloyal to have a cultural identity from the Old Country. That would require every Italian American (for example) to deport themselves unless they've expunged every whiff of an accent or odd (to Americans) custom. Taken literally, he's saying that you must become "American enough" ("in every facet" ) to be worthy. And then he makes this somehow synonymous with "loyalty". I suspect he was playing to the crowds of his time.

While it does personally annoy me that many immigrants make no effort to learn proficient English and seem weirdly intent on not embracing the behavioral and cultural norms of their new country, and an argument can be made that such an attitude can be rather less than fully respectful and humble ... it IS hard for people to give up the visceral identity that they acquired as children. What ends up happening is that it takes a couple of generations for assimilation to fully happen in a family, and in some ways that is understandable and very human and should not be automatically judged. Particularly when you consider that not every immigrant comes here totally out of free choice, they may be escaping some form of national suffering, persecution, famine or whatever, and have had an identity ripped from them that they experience as a loss.

It's easy for me to sit here and say that if I immigrated to France I would be all over fluency and clarity in French and a good command of French culture, but faced with the reality at my age of adopting a new country, some aspects of which would inevitably grate on me, it might fall to my children to get their brains fully around it.

I disagree with the statement. I think America has "room" for more than one language. That's just a fact. People speak many different languages in America. And there must also be room for more than one loyalty. That's just what freedom of speech is about. You can be an American citizen and still hold some loyalty to another country. As long as there is no war with that other country I see no problem there. For instance, you can be proud of your Italian roots even as an American.
Roosevelt said this statement in a time where the world was less connected and nations were much much more important than today. War wasn't something that was done only to underdeveloped countries but between the richest nations. Loyalty was more vital back then. So I understand where he was coming from but times have changed.
So do I mean nobody has to assimilate into a culture they emigrate to? I think they only thing you have to do is follow the laws of the country. If your culture supports child marriages you have to stop that in countries that don't allow that. Other than that? Live and let live.