Yeah, the "WWII rocket scientists" argument isn't really one you can extend to regular immigration, except to say that you benefit from bringing people in with complex skills that your country lacks.
I think the Nazi rocket scientists signed up with America mainly because the alternative was to be tried as war criminals at Nuremberg. Those on the Russian side weren't even given that option.
Statistically, Japan is in "Oh shit, sweet mother of god, we are in serious SHIT!!" territory there. Historical numbers indicate imminent cultural collapse, but then again, old data is for a world with much higher mortality rates.
They at least see it coming and are trying to prepare for it. Unfortunately, Japan is one of the most insular countries in the world as far as immigration goes. Not that they won't take immigrants, they're just highly resistant to it. They're incredibly open to most everything else though.
Countries go through high-fertility, to low-fertility periods as the economy grows, basically they go from "men work/women have babies" to "both men and women work". And conventionally, this is seen as the "end game" of development.
But some demographers are saying there's evidence that there's another phase
after that one: basically productivity becomes so high that women can choose whether to work or have babies again, and there's a big uptick in fertility rate again. See places like Sweden for example. In a post-automation future it might be that people find they've got more time to raise kids properly, they're not couples both with 60 hour a week jobs and tied down with massive bills.