Our bodies are not well-designed, they are full of legacy. The whole idea of moving via contracting meat is... strange. For real, lower half of me exists only to move the upper half. That's crazy.
As a social species, in the development of humans it's not only the biological structures that are important (the body itself), but the informational, social structures. A tribe which works well socially (helping each other, keeping together etc.) will survive much better and longer than a tribe of individualists, and yet even better than individuals living on their own. A big part of actually keeping a tribe structure alive is the knowledge than elders constantly pass on to the new generation. And that does not stop after the genes are passed.
Additionally, there seems to be some (deeper, philosophical) knowledge that is very hard to acquire before a certain age. It just takes that long to collect enough experience to be able to connect certain high level concepts. It's very likely that a tribe with longer-living monkeys (metaphorically) would have more of these wise elders that have acquired the knowledge and could pass it on better to the rest, than a tribe with monkeys that would die off fast after making babies.
But this still would only mean that the evolution would not kill you off directly after passing on the genes, but instead some time afterwards, after you've already learned all important things about life and were able to quickly teach them to the new generation. It would still kill you off either way.
A plausible explanation of aging is for example the evolutionary shadow: From the perspective of your genes you don't matter (much) after you've successfully passed on you genes to the next generation. Thus there is no evolutionary pressure for longevity. The programmed cell decay of aging might be a way of multicellular organisms to fight cancer.