Lets analyze the scenario where individuals don't die by aging. So we have some members of the species from generation 1 who grow up, then reproduce. In generation 2, there are some that have mutations that make them more suitable for the environment and some that make them less suitable. The less suitable ones are outcompeted and don't survive, the normal ones mostly group up to survive, and the ones with beneficial mutations outcompete all the others, leading to the deaths of some of the ones with the "original" genes, and procreate. This then happens again and again with beneficial mutations and they eventually becoming ubiquitous in the population.
There's just no way that an individual member of the species with a mutation that gives it a definite lifespan has a fitness advantage over the others, all else being equal. What's more likely is that immortality is not necessary to pass on genes, so it simply never developed because it would involve many complex systems, the individual components of which would not have conferred enough of a fitness advantage on their own to become universal in a population. And since the natural environment is harsh and predatorial, most animals don't die of old age anyway, which again limits the usefulness of an unlimited potential lifespan.