So the immigrants end up with the same birth rate and then we have to bring in more immigrants. So there would have to be a constant source of excess people somewhere else for this to be sustainable. So it gets to a point where I wonder if this is really the plan.
You're assuming the birthrate can be "fixed". Japan has an even worse demographic problem. They've tried "fixing" it and nothing much works. Similar for South Korea. Japan would certainly benefit from opening up to more immigration. Countries that can attract more immigration are going to benefit economically as the birthrates continue to decline globally. Why not be attractive to immigrants instead of wanting to repel them?
Not to mention how unsustainable it is anyways, what happens a few years down the line when the immigrants stop having kids? There isn't an infinite ocean of people you can import on a whim, and definitely not forever.