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Despite common belief, accumulation of molecular damage does not play a key role in aging. Still, cancer (an age-related disease) is initiated by molecular damage. Cancer and aging share a lot in common including the activation of the TOR pathway. But the role of molecular damage distinguishes cancer and aging. Furthermore, an analysis of the role of both damage and aging in cancer argues against "a decline, caused by accumulation of molecular damage" as a cause of aging. I also discuss how random molecular damage, via rounds of multiplication and selection, brings about non-random hallmarks of cancer.
Mikhail V. Blagosklonny (Sat,) studied this question.