Everyone dies of something.
Go and survey all the apes in the wild. Everything from murder to falling out of trees, to predators, to falling-out-of-trees-while-fleeing-predators.
Most animals DO NOT die of old-age. That's a very human-centric view.
Getting eaten is visible on the fossils. Disease is often visible too, or suspected only because there are no other injuries (which is suspicious in itself). Even Tutankhamen is thought to have had several fractures when he died and he was only a boy.
For a tree-dwelling species, dying from falling out of a tree is right up there. Once you slip once, whether learning toddler, careless adolescent or fleeing adult, you break bones that are a) visible on your skeleton and b) crippling to your ability to survive.
No antibiotics. No way to monitor or stem blood loss (especially internally). No knowledge to heal the bone. No painkillers. Can't keep up with the pack. You're dead. Hell, you could have just picked the rotten branch and by the time your weight was on it, it was too late to do anything.
Watch a cat. The most graceful of animals. Sure-footed. Sleek. Can land on their feet from stories up. Able to leap up and down trees at stupendous speeds with little or no warning, dive over obstacles, sprint faster than you ever could.
In the last year, from three cats in my house, two have fallen off a windowsill more times than I care to mention, one got trapped in a catflap (by backing out of it while half-way, requiring human intervention because it just kept pulling on it while its tail was caught in the flap the wrong way to escape the flap), one got stuck in a tree, one has a supreme deathwish where sitting in front of moving cars is concerned and only saved by driver prudence (i.e. me), one has come back with bloodied paws on more than one occasion (believed to be from a bad jump down from said tree again, onto sharp ground!), and that's not counting modern hazards, predators, actions made under panic, running between human legs on stairs, etc. for a domestic cat roaming a small garden territory.
I've actually just watched one fall off a sofa because it was sitting on the back of it, went to rub against my hand, misjudged it, and fell to the floor. It shook it off, but it completely messed up a simple action. And this was a young cat, not a kitten or something too-old-to-survive.
It's like saying a professional juggler never drops his balls, or that a professional acrobat never misses a leap. Ask them. They ALL do. They just don't always do it every show. But put enough shows on (i.e. climb enough trees) and it will happen eventually.
Few animals EVER reached old age, unless they were impregnable or zero-risk animals (e.g. tortoises, elephants until humans came along - slow, ploddering, no jumping, etc). Almost none of the hunter-cats ever really get to old-age because they all die of simple injury or infection of injury. There's not much to challenge an old established-pride lion, but the simplest of slips on a rock will kill him.