The real question is why they cannot produce more people within their own country.
However they have a right to open their country to immigration if that makes sense for them.
“Experts say the reasons for these demographic shifts across the region include demanding work cultures, stagnating wages, rising costs of living, changing attitudes toward marriage and gender equality, and rising disillusionment among younger generations.”
Link: https://www.cnn.com/2023/12/15/asia/south-korea-to-see-population-plummet-intl-hnk/index.html
It’s really quite dire, as it happened so fast and the fertility rate took a nose dive:
“Over the last 60 years, South Korea has undergone the most rapid fertility decline in recorded human history. In 1960, the nation’s total fertility rate — the number of children, on average, that a woman has during her reproductive years — stood at just under six children per woman. In 2022, that figure was 0.78.”
Link: -fertility-rate-in-the-world-and-that-doesnt-bode-well-for-its-economy.html#:~:text=Over%20the%20last%2060%20years,2022%2C%20that%20figure%20was%200.78.
They are faced with a tiny working age population supporting a larger population of elderly.