You don't need exact stability though, having a fertility rate just above or just below replacement rate in periods wouldn't be too big of a deal.
But now we are facing a crisis. For instance the next South Korean generation is set to be half of the current one. And fertility declined strongly there over the past 10 years. Many countries are now at levels SK had ten years ago (well below the replacement rate by higher) and if the trend continue they may end up in the same situation as SK in a few years.
Halving your population over one generation is a catastrophic event.
But now we are facing a crisis. For instance the next South Korean generation is set to be half of the current one. And fertility declined strongly there over the past 10 years. Many countries are now at levels SK had ten years ago (well below the replacement rate by higher) and if the trend continue they may end up in the same situation as SK in a few years.
Halving your population over one generation is a catastrophic event.