Senescence, immortality and evolution?
You will find here a very good overview of the main hypotheses that explain senescence.

I am not sure I will answer your question but here are some reactions to your post..

What you say has to do with species selection. Selection selects between various objects/units that differ. These objects may be genes, individuals, populations, species, etc… In other words, there are various level of selection or different units on which selection acts. Wiki

Generally speaking, it is often assumed better to think in terms of selection acting on alleles (gene variants) or on individuals. Therefore, we tend to think that selection acting on genes or on individuals is more effective than selection acting on species. However, selection acts on species and we should not necessarily discard explanation using the concept of species selection.

What you say, if I am not mistaken, if we consider a set of species that are either mortal or immortal, we would predict that selection would benefit the mortal ones. Stating differently, mortal species have some advantage over immortal species that allow them to not get extinct or eventually to speciate more.

However, I don't quite understand the reasons why immortality would be deleterious to species. You say that it leads to overpopulation. So would you expect that a population is lead to overpopulation due to immortality and suddenly disappear because it is overcrowded. It doesn't sound like a well formulated argument to me. You could however argue that mortal species have a much greater individual turnover allowing them to adapt faster to environmental change than would immortal species.