Easy. Teddy defined it:
Someone who does not...
at heart feel more sympathy with Europeans (or other nation/culture) of that nationality, than with the other citizens of the American Republic
And someone who adheres to our laws and not some other cultural/religious system of law.
Not so easy, actually.
Teddy didn't 'define it'. The term 'hyphenated American' was a popular buzzword around the end of the 19th and beginning of the 20th Centuries in the USA. It had a lot of meanings, and Teddy just defined for himself a particular meaning.
It also meant (according the Anglo American Society), anyone who was Catholic, Irish, or God help us, Catholic and Irish. Fact. It also meant (to them) anyone who was "anti-expansion, anti-imperialist, anti-English, or pro-Boer. How's them apples?
It also meant, according to the Deseret, the magazine of the Church of Latter Day Saints, anyone who was a member of the British-American Society and were currently (according to them) attempting to pin crimes on Irish-American citizens and the government for having recently appointed a British citizen to the post of Ambassador to Chile after having replaced a German citizen who had taken American citizenship.
The book "Slang and Its Analogues," published in 1893, defined 'Hyphenated American' as meaning: "A naturalized citizen, such as German-Americans, Irish-Americans, and the like."
The magazine "Public Opinion" by the Progressive Conservative Party of Canada, had something to say about the situation in the USA in 1899:
We are told that prominent German Republicans in Chicago are advising President McKinley that if the present policy of expansion is persisted in the Republican party will lose the German vote in 1900. This country is not disposed to listen to the advice of Germans, Irishmen, Englishmen, Spaniards, Italians, or people of whatsoever race. If the German-Americans choose to raise such an issue as this; to challenge the American in his own home on a question of loyalty to America, or loyalty to Germany and German ideals, we believe America is ready for the issue, and we believe the German will for the first time in his life find out how many Americans there are in this country. Any race within the republic that acts as a unit is a danger to the republic. It becomes an imperium in imperio. In our great strength and our great love for freedom we have perhaps been too careless of this in the past. Captain Coghlan, from his experience abroad, warned us of the danger of these "European colonies," as he stigmatized the hyphenated American citizens, telling us that surely they would do us harm if we allowed them their way. We might take a lesson of Germany itself, which so savagely suppresses ail languages but the German language; so fiercely proscribes all customs but German customs; so insistently Germanizes everything under its control. We do not know of anything that would cause a greater rally to President McKinley and his policy than such threats as these that we have been discussing.
We might take a note from history and discover that despite nothing having been done in the early 1900's to make illegal being 'German-American' or forcing English to be spoken in the home, etc, as the writer demanded, we seem to have remained un-taken-over by Germany.
And of course, I must note that even your own description of what President Roosevelt meant when he said:
at heart feel more sympathy with Europeans (or other nation/culture) of that nationality, than with the other citizens of the American Republic
...is not how the quote is used today, is it? It is not quoted and then explained to mean just that - people whose loyalties lay elsewhere. No, it is used as a general pejorative for anyone who has an cultural identification at all. If a person's ancestry is Irish or German or Mexican or Polish or Iraqi and they are proud of it and refer to themselves as an Irish-American (or etc), then they are hammered down and told that NO, they must reject such things and be a PURE AMERICAN and NOTHING ELSE. Even if all their loyalty lies with the USA, they are just demonstrating pride in their familial origin, that hoary old quote from Roosevelt is trotted out as if that's what he meant when he said it. Of course, that is not what he meant when he said it, as you have made clear yourself.
I agree with Roosevelt; people who take US citizenship should be loyal to the USA and to no other country! But calling themselves
'Mexican-Americans' is not the same as saying they are loyal to Mexico and not the USA. Some would pretend it does.
Taking pride in their cultural heritage, preserving their language and traditions, keeping their own cuisine, customs, and values, none of these things preclude them from being loyal US citizens and NOTHING ELSE. But some would pretend that it does. Teddy never said that. People twist his words to make it seem that he said that.
And of course, it is selectively applied - then and now. Then, it was applied to Irish-Americans (by Anglo-Americans, hahahaha how ironic) and to German-Americans (by Canadians!). Now it is applied to Mexican-Americans and Muslims who are American citizens or legal residents - by others who have no problem being of Irish heritage or German heritage or Polish heritage or no heritage in particular.
I'm of Welsh extraction myself. I feel no loyalty to Wales, but I have a Welsh flag at home. I think it's neat. I think Wales is a cool country. I like it. I don't speak Welsh, but my ancestors did. I guess they were
'hyphenated Americans', eh? Bad, bad, people. I should be ashamed of them. And if they had managed to keep some of their traditions or language intact and pass it down to me, why, we'd all be disloyal, treacherous dangers to the USA who ought to be savagely suppressed for the good of the nation. RIGHT?
And of course, I have to complete by asking once again - assuming you are right and I am wrong and being a hyphenated American is a bad thing - what is your proposed solution?
I'm still waiting to hear the solution.