In an overall well-written and thoughtful article, this claim stands out to me as being particularly lacking in empirical support. Of course logically it is true inasmuch as people are people. But practically is there actually any support for it working on a scale large enough to make any substantive difference in places like Japan?

The problem is that when people move they leave their country behind but not (entirely) their culture. Even if there is zero prejudice against people simply because of their different ancestry, this can cause difficulties in the destination country. Culture is very important to human well-being, and older people (empirically) have a particularly hard time adjusting to cultural changes. And you can't bring people from other places in as babies, whereupon they'd organically pick up the new culture, because birth rates are low not because pregnancy is so onerous (though obviously that plays some role), but rather because people do not feel they can raise a child.

Possibly the best example of successful large-scale immigration is of Hispanics to the United States, into a country of immigrants where the culture is already pre-adapted to that kind of thing, and it still causes strife, and has only raised the effective birth rate by 0.1 during the most rapid period of expansion (1990s/2000s). In Europe, immigration has led to significant social strife in multiple countries--witness the issues in France, for instance, whose immigration levels correspond to a roughly 0.18 increase in the birth rate.

The societal changes necessary to handle large numbers of people with considerably different culture and values are far smaller than the changes necessary to make it rewarding to have more children (still below replacement value, but not so much below replacement), so I'm rather puzzled by your claim here. It seems more aspirational than evidence-based.