But I'm sure you'd agree that if a culture changes 'quickly enough', it's not really changing it but basically deleting it. For instance I'm sure if you squint hard enough today, maybe there's some Native American influence on American culture, but for the most part we just destroyed it. And I don't think it was the violence, so much as the raw numbers. Something similar happened to the Maori in New Zealand without anywhere near the scale of violence, just lots of immigration and lots of babies.
So how fast will fertility change things? A fertility rate of 1 means each generation decreases in size by about 50%, but that's compounding. The length of a generation is proportional to the window of human fertility, which is about 20 years. So in an average human lifetime we'll each get to see around 4 generations play out. With a 50% compounding decline that means a generational decline in size of 94% in a human lifetime.
So if you want to try to deal with fertility declines with immigration, you're basically going to be replacing the entire population in extremely rapid order. So I don't think we really need to debate what 'quickly enough' means because it's way slower than what fertility collapse is going to do. And this is all far more true in places like Japan where you, out of necessity, are speaking of bringing in immigrants who the locals can't comfortably communicate with and who come from radically different cultures. So basically, from my perspective, I think you're speaking of having your cake and eating it too.