How come that chain hasn't failed?
Basically, you're just rephrasing the "aging through accumulation of errors" hypothesis. The problem is, errors should accumulate along the chain mentioned above, too. Yet they don't.
An individual human body has error correcting mechanisms that do not benefit from being renewed every cycle- they deteriorate over time, Individuals suffer from problems larger systems don't - a few cancerous cells can poison the whole individual, but a few cancerous individuals are likely to die with few offspring, and their offspring are also more likely to die with few offspring. If an individual is left with only a few correctly functioning cells, they have no recourse for rebuilding themself.
Species get a constant influx of fresh water, as it were, and individuals have to make do with a set quantity.
(There's several points of 'selection' for sex cells- error free ones are, loosely speaking, more likely to successfully fertilize/get fertilized, and more likely to create viable offspring that themselves reproduce.)
Edit: It's like the difference between being able to select the top 10% and selecting those with a score above 90%. The chain gets to "select" only those that perform above a set threshold, but the organism has to play out every hand it's dealt.