This is the actual answer. Immigration is fine, but you have a sick society if it doesn't feel like it can risk having children. Immigrant aren't magic and after one generation they will succumb to the same problems of that sick society.
They'll do the Japanese thing and allow slightly more Filipino nurses to immigrate to take care of all their geriatrics and nothing else.
Japan and South Korea are gonna ride their xenophobia until the wheels fall off.
Read a book recently about demographic 'replacement rates' and how low it was in South Korea and Japan (as well as a LOT of other countries, including in Europe).
Author pointed at Canada as a success story of reversing the trend. Offered an example of how immigrants, especially those who could start businesses or were educated could help with the demographic issue without impacting the economy.
But you have to first get around the xenophobia issue. A minister standing up and pulling the proverbial fire alarm is a first step.