That assumes birthrates elsewhere are high enough to sustain that, but they're declining everywhere. Immigration is basically just exporting the low birthrate problem to someplace less equipped to deal with it (which is par for the course for rich, western countries).
I expect we'll start to see emigration controls in the not too distant future.
I think China's approach would become the blueprint for third world countries eventually dealing with this.
Unfortunately China doesn't seem to have an approach yet, other than doubling down on more and more authoritarian measures to lock down the economy.
IIRC, they've only tried carrots so far, which hasn't worked. My bet is they'll turn to sticks, sooner rather than later, and have a lot of experience with them due to the one child policy.
I wouldn't be surprised if they start to fine the childless and/or limit childless women's advancement opportunities at work.
I do not agree with emigration controls. At least not in the next 50 years!