I agree; that line of thinking is only valid if aging is considered to be "beyond the ability to reproduce." If the organisms can still reproduce at the same rate, they are not too old to contribute to natural selection. However, a cumulative distribution over the number of offspring created by age for a particular fitness model will regardless overshadow reproduction in old age with the aggressive reproductive periods during youth, and there are likely diminished evolutionary returns for keeping an organism around longer.
On an ecosystem/gaia scale, there is also the argument that organisms that never die will break the ecosystem by failing to return nutrients to the nutrient cycle. However, this is not necessarily an argument against biological immortality, as there are plenty of other ways to die (starvation, predation, etc.)
To expand on your second paragraph, I would say it's not argument at all because it implies purpose. There is nothing to stop the ecosystem from breaking.
If there was a species that "fixed" aging and continued to reproduce, it would be so successful it would take over and break the ecosystem. This was never observed, not because of "purpose", but most likely because there is entropy limit to extending lifespan (mentioned in the end of article).
On an ecosystem/gaia scale, there is also the argument that organisms that never die will break the ecosystem by failing to return nutrients to the nutrient cycle. However, this is not necessarily an argument against biological immortality, as there are plenty of other ways to die (starvation, predation, etc.)