Alternatively they could choose to not have significant immigration into their societies, and instead deal with the low birthrate problem directly.
How would you do that?
Apart from that that, there is no inherent problem with immigration per se. Skilled immigrants who identify with a liberal society are an asset. The issue is how to make sure that people who want to stay here integrate well into society instead of creating isolated parallel sub-societies, and how to reject those who don't.
But clearly, the solution is to take in millions of unskilled refugees from failed nations!
Taking in refugees is not meant to be a solution for any domestic problem. You are conflating immigration with asylum. Both are different problems with need for different solutions.