Suppose they choose to "solve" it with immigration - what will happen? It will enable their current fertility-destroying system to persist, shrinking their population indefinitely, while supplementing it with immigrants.
Social-economic systems that cause below-replacement fertility are in this way much like parasites, feeding off the host nation. Without immigration, eventually this will cause enough of a crisis, or a high enough fever, to continue the analogy, that will allow the nation to shake off the parasite and change their system to something sustainable.
But immigration functions as anesthetic and immunosuppressant - keeping the host population just comfortable enough that they go gentle into that good night, and giving the parasite economic system a new host to feed on.