I think, if you are concerned with the cultural shift, you can give immigrants temporary term VISAs and make it clear their stay is going to be strictly temporary. That was supposed to be how the Gastarbeiter system works. Thing is, when you have already on boarded a foreign worker and have had them working for you for 2 years already, you don't want to let them go and replace them with a fresh foreign worker who you have to retrain.
My girlfriend asked me once in a Restaurant (Cafe Norden in Kopenhagen) why I was ordering in English. Then, I spent the complete evening ordering everything in Danish, to always receive answers in English. And this nearly everywhere (not my bakery, the girls there could not speak a word of English, this was an exception). This was in the early 2000's. This never happened to me in all the other countries I have been living in. A Danish colleague simply told me that he does not like to listen to broken Danish, better switch to English.
Still have a lot of friends in Denmark, integration there is not easy, even for highly qualified people from the EU.
In the US, Italians, Irish, and Germans, as just a few examples, all were treated as unassmilated aliens. People say the same things, generation after generation about newcomers. If you moved to a country with a different language and culture, you might make an effort to learn the language but you would still think, read, etc. in your native language. Your kids would know both languages; their kids would of course speak the language of their surroundings.
What solves these problems is a fundamental belief in freedom for all. People don't need to speak your language to be your equal and be just as human, with the same rights as you.
Here's Benjamin Franklin (Palatine refers to the Holy Roman Empire or Germanic regions, depending on the source):
"[W]hy should the Palatine Boors be suffered to swarm into our Settlements, and by herding together establish their Language and Manners to the Exclusion of ours? Why should Pennsylvania, founded by the English, become a Colony of Aliens, who will shortly be so numerous as to Germanize us instead of our Anglifying them, and will never adopt our Language or Customs, any more than they can acquire our Complexion.
Which leads me to add one Remark: That the Number of purely white People in the World is proportionably very small. All Africa is black or tawny. Asia chiefly tawny. America (exclusive of the new Comers) wholly so. And in Europe, the Spaniards, Italians, French, Russians and Swedes, are generally of what we call a swarthy Complexion; as are the Germans also, the Saxons only excepted, who with the English, make the principal Body of White People on the Face of the Earth. I could wish their Numbers were increased. And while we are, as I may call it, Scouring our Planet, by clearing America of Woods, and so making this Side of our Globe reflect a brighter Light to the Eyes of Inhabitants in Mars or Venus, why should we in the Sight of Superior Beings, darken its People? why increase the Sons of Africa, by Planting them in America, where we have so fair an Opportunity, by excluding all Blacks and Tawneys, of increasing the lovely White and Red? But perhaps I am partial to the Complexion of my Country, for such Kind of Partiality is natural to Mankind."
https://www.theatlantic.com/politics/archive/2008/02/swarthy...
[1] https://www.pure.ed.ac.uk/ws/portalfiles/portal/35314239/sre...
You can look at any country in western europe to see how it's not.
In practice that rarely goes as intended. The first political party to say, "We changed our minds, you can all have citizenship," gets to secure power with an influx of loyal voters.
Before: 5 million illegals or immigrants.
Party A passes law making them all citizens.
After: Party A always wins because large portion of the new 5 million vote for Party A
Conversely, if this hypothetical party had a secret agenda to pass the law when they get elected, but they kept this a secret until then... first of all they will probably lose of voters in the next election. Also, they might find out that former immigrants do not automatically reward this behaviour in terms of votes.
I do not find this scenario particularly realistic.
In America, we discovered basically by accident[0] how to bring in large groups of people and have them be economically and socially productive. Specifically, we had very generous family reunification programs, which outsourced the question of "what immigrants do we admit" to immigrants who had already successfully integrated. Effectively America became an invite system. The end result is that, for basically any background, every American city has a large and established immigrant population from that country for new immigrants to fall back on.
Other Western countries (e.g. western Europe, Canada, the UK, Australia, South Korea, and Japan) didn't do this. Instead, they treated immigration as a transaction: you can't come in unless you have some immediately beneficial purpose, and permanent residency is going to be an even higher bar still. What this gives you is immigration systems that select entirely for disaffected, college-educated workaholic youths with no connections to the local city. To put it in the words of Canadian Prime Minister Mark Carney:
"America is a melting pot. Canada is a mosaic. In the United States, they don't recognize differences[1]. They don't recognize First Nations. And there will never be the right to the French language."
These words are telling, in ways Carney probably didn't intend. He is proudly admitting that the thing that sets Canada apart from America - the thing that they should fight with military force against a wannabe dictator with annexation dreams - is the fact that they can't integrate people worth a damn.
[0] Family reunification visas were originally intended to deliberately preserve racist prejudices in the US immigration system. This backfired comically.
[1] ...has this guy never heard the phrase 'African-American' before?