Complaining about one-kid-a-century might make sense if you were, say, King Thranduil in the Second Age and your young people are constantly being eaten by spiders, but it makes no sense whatsoever if you are Ingwe of the Vanyar and only one of your people has died ever in all history. (Obviously those are extreme cases, and Tolkien actually does a reasonably good job with it, but a lot of the people building off of Tolkien have ...not.)
Basically it's back to you want your birth rate to basically equal your death rate in order to be stable. If you want to think about it generationally, which is easier than thinking in the large scale for me when I'm storybuilding, in a particular age cohort of immortals the percent of that cohort who have died so far should (very roughly!) equal the the percent of that cohort who have had a kid so far*. If it is significantly less, then you can start worrying about your people dying out. If it is significantly more, you should probably be sending more people out to fight the giant spiders.
*Although this only applies if your deaths and births are evenly distributed by age. If your death rate is very skewed by age - say if most people who die don't make it to their first century - you can probably give their agemates a couple more centuries to catch up with the birthing, but once the survivors have made it past the high-death-rate age, it would still pretty much apply. And if you can only have kids in your first century then the young people would have to get well ahead on the birthing.)
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Complaining about one-kid-a-century might make sense if you were, say, King Thranduil in the Second Age and your young people are constantly being eaten by spiders, but it makes no sense whatsoever if you are Ingwe of the Vanyar and only one of your people has died ever in all history. (Obviously those are extreme cases, and Tolkien actually does a reasonably good job with it, but a lot of the people building off of Tolkien have ...not.)
Basically it's back to you want your birth rate to basically equal your death rate in order to be stable. If you want to think about it generationally, which is easier than thinking in the large scale for me when I'm storybuilding, in a particular age cohort of immortals the percent of that cohort who have died so far should (very roughly!) equal the the percent of that cohort who have had a kid so far*. If it is significantly less, then you can start worrying about your people dying out. If it is significantly more, you should probably be sending more people out to fight the giant spiders.
*Although this only applies if your deaths and births are evenly distributed by age. If your death rate is very skewed by age - say if most people who die don't make it to their first century - you can probably give their agemates a couple more centuries to catch up with the birthing, but once the survivors have made it past the high-death-rate age, it would still pretty much apply. And if you can only have kids in your first century then the young people would have to get well ahead on the birthing.)