It's been awhile since I've read up on cellular biology, but wasn't the theory for the aging problem related to the telomeres not being 100% replicated during cell division?
That is one factor that apparently has a significant input, but it is not the only posited mechanism.
Comparative studies of other organisms do show examples where longer telomeres or better maintenance of them do pay off: tortoises and long-lived reptiles are an example and cancer cells often have rampant production of enzymes that rebuild them (whether that is a cause or effect of what makes them so long lived, I don't know).
Significant extensions of lifespan are possible with other manipulations, at least for creatures like mice, nematodes, and fruit flies.
Caloric restriction, selective deletion of certain genes, better genetic upkeep, proteins that help mitochondria repair themselves, are other ways to get long-lived animals.
It's probably a multi-stage question with differing thresholds.
Perhaps removing in-built restrictions allows a creature to reach the age where telomeres become a problem, and perhaps alleviating the threshold for telomeres just leaves time for other things like mitochondrial degradation and accumulating mutations to come to the fore.