Interpreting this post as a kind of argument, I have the following to say:

First, you seem to be conflating death with aging. For instance, in your first several sections, you present an argument for why death is necessary. Let's say we grant you that: death is necessary. But then you go on to say that "This is to cut down the amount of competition for resources between parents and the next generation." Well here's a way to cut down on the competition: have the kids die.

I mean, there are organisms that, for all we know, are immortal. Whether you (a) keep on living, or (b) have two children that each share 50% of your DNA and then die, or (c) have one child that shares 100% of your DNA and then die, is all the same. So why pick the second way?

In other words, you haven't explained aging. You have explained death. But death can be achieved in a number of ways (a, b, c), parents dying (b) being just one of them.

Secondly, I think you may have put the cart before the horse. Animals reproduce more when they die more, they don't die more because they reproduce more. In other words, if elephants had a very successful predator that killed most of their offspring, or if they had rich food sources, then elephants would adjust by producing more offspring to offset the killing of the offspring by the predator, or to take advantage of the rich food sources.

Third, though this may be the second point reversed, if resources are scant, one possible adaptation is for nature to make the rats/bacteria/fish to produce less offspring, so there won't be as much competition between the offspring.

Fourth, you're treating death as some sort of force of nature, keeping numbers in check, but it's unclear how it works. To me it's just self-evidently true that no organism can grow beyond the environment's ability to sustain that growth. There's no need to bring death into the picture, it's more like a metaphor.

Simply, from a Darwinian perspective, the purpose of an organism is to be born, reproduce some offspring that are slight variations of themselves and then exit the scene as quickly as possible.
This is to cut down the amount of competition for resources between parents and the next generation. The faster the cycle can repeat itself, the faster the environment will select for better adaptions and the species will evolve.

I think this is false. It sounds like teleology. I don't think creatures aim to evolve. I don't think mutations happen with foresight, or purpose. If the environment has been stable for a long time, cells that tend to make errors in copying themselves will disappear, since by definition all mutations will be less adaptive (since the environment is stable and we assume adaptation has reached a saturation point). So organisms will tend to produce very faithful versions of themselves. If on the other hand the environment changes often, then either cells making more errors will be favored, or else the same number of errors but more offspring (as in bacteria). But there's no need to talk of purposes: it no more serves an organism to produce variations of itself than it serves a warbler to raise a cuckoo.