Yet the fact of British settlers importing some French or Catholic ideas is VERY different from them BECOMING French. Otherwise New England would the extension of Quebec.
The structure of our entire country [...] is also different from that of the Romans, btw. Anybody can become an American by accepting our ideas.
What do you mean by acceptance of ideas? When Germanic immigrants accepted Roman dress, titles and citizenship, said and thought that they "accept Roman ideas", and styled themselves as Romans, did they become Roman? It is what everybody liked to think at that time. Yet when the ethnic/cultural core or Latin weakened Rome changed meaning.
Despite that the late Roman experiment of melting pot and absorption of mass immigration and of other ethnic/cultural lands WAS MORE SUCCESSFULL than we tend to think about. even with the Latin core being marginalized.
The FORGOTTEN and quite successful experiment was the Eastern Rome - Byzantine Empire which combined Roman statecraft/statehood with Greek culture and Eastern spirituality (Christianity). It lasted THOUSAND years.
The ethnic core group which replaced Latin place of leadership - the Greeks, accepted Roman identity and ideas of state/government. Latin was the official language for a long time in the New Rome by the Bosphorus, the Roman Law (perfected later by Emperor Justinian), Roman institutions and titles were preserved.
But there are two key elements that might be missing in other attempts to emulate Byzantine experiment. First Greeks the people among whom it was implemented were the most talented, creative, lively and curious that walked on the surface of the earth. Second the Roman state in its original Latin form was already imbued with Greek ideas making it easier to perform this transplant.
And what is telling after Eastern Rome was destroyed by the Turks and after centuries of Muslim yoke the people who emerged from ruins are and consider themselves Greek! (Even if they retain some elements of adopted Roman identity.)
Consider what happened to the Caledonians when the Scots moved in.
Sorry, but if you think becoming American is the same as becoming a WASP, there's really nothing more to say. This nonsense popped up with the Know-Nothings (the modern heirs of whom seem to populate FR), was disproved then and shown to be the mere bigotry it was, and I think the modern anti-Hispanic hysteria will meet the same fate.
If you're referring to the modern country of Greece, they have a birthrate of about 1.5 and a growing number of mosques in the country. European Turkey Redux, anyone?