Earlier quoted context omitted.
Moles don't stop growing primarily because of telomere length but because the human genome has effective methods of controlling cancer. You get cancer something like 4 times a day but the cells are terminated. (It is possible telomeres play some part in this, but I have never seen it proposed.) What you are likely trying to broach is the question "do organisms evolve to die?" the answer to which is "yes." This was or…
Imagine humans who can't die from old age: how many would choose to do anything dangerous? Would you risk your life crossing a road? Working on a construction site?
Suppose we stop aging - this may be great, as we could live at our physical peak until death and the life would be much much longer on average. But at some point, maybe 100 - 200 y/o depending on individual, people's mind may just break.
To truly achieve immortality, we would have to probably reverse engineer the brain and gain complete understand of how it works. Even if we could copy brains atom by atom, or upload it to virtual world, without understanding of how to assemble a brain, this will be useless. A virtual brain, which would not be enhanced in some way, would age exactly like a real world one.
So probably the scenario where people can't die of old age is almost impossible, because that requires technology more advanced than even the ability to backup/restore a mind - which would allow people to die in accident and be restored - although most certainly there would be laws to regulate it, unless we have practically infinite space / land for people to live.
Sorry for going all Sci-Fi on this, I watched the 6th Day yesterday. The antagonist thought he could be immortal because he had a machine that could clone (or rather copy) a person including the brain. He had the ability to get himself resurrected if he died, but he probably wouldn't be able to live even a thousand years, unless he also had developed the technology to prevent data corruption in his brain.
Unless maybe I'm completely wrong and the brains are so flexible they would just work, but there's no way to test it, is it.