The whole immigrants who don't integrate seems to be a constant issue in every developed country across the globe, with the exception of maybe Japan, who is xenophobic enough that you wouldn't want to try to become Japanese anyway.
Canada, England, France, and the US, to name a few, seem to have done it wrong considering how immigration is a constant complaint and weaponized topic in their politics, but likewise Japan has too, just on the other end of the spectrum.
I'm unsure who does it well.
> The whole immigrants who don't integrate seems to be a constant issue in every developed country across the globe
I can't speak for the other countries, but in the US it's almost entirely an agenda being pushed. When I hear people say this, it's not because of any experience they've had, but just a repetition of talking points. Virtually none of them had a negative story to tell. I've heard far, far worse stories in some European countries.
Immigration is a hot topic in every society. You can hear people in Arizona kvetching about the Californians that move there.
Immigration isn't a binary decision for a state. Its a complex set of policy decisions. What is the state of the local economy? What skills do the new immigrants have? How much experience does the local population have with the incoming culture? What is the culture of the incoming migrants? All of these and many other things comes into play when evaluating or deciding upon immigration policies. For example, taking in Catholics from northern Mexico is nothing like taking in people from the tribals of Pakistan. Most European (quite naive) immigration policies from 10 years ago seemed so poorly thought out that they were doomed to fail. The swing the other way seems guaranteed based upon how bad those previous policies were.
PS The US does it well in general but there were periods of madness recently. The political discourse of many still shows extreme madness is still possible.
PPS The very wealthy are normally the main beneficiaries of immigration and undocumented immigration just creates an underclass with few to no legal protections.