I don't know anything about trees around there, maybe they're really short-lived? For forests around here, it's a gross overestimate.
Also, I bet it's not just the tree you're sleeping under that poses a risk, but also other trees in the vicinity that might fall on you. In a forest there are probably a bunch of trees "in range".
All of that said, I've often camped and slept in forests, as have many of my friends, and I've never heard of anyone being killed or injured by a falling tree, or ever heard or seen any "don't sleep under a tree, it might kill you"-advice, so I don't know...
I would still guess that the number is wrong, I suspect that trees have a longer life on average from becoming what we would consider "big" until they fall over. But it's still an important detail.
The logic in the story is BS. First, you would never, ever get into a car with that logic. Second, trees just aren't that ephemeral. People who live in a forest would be very aware of when they do or don't fall (or more problematically, drop large branches.) It's not as straightforward as just avoiding storms or dead-looking trees. Sustained wet weather, especially after a period of dry weather, is a common cause. As is the opposite for some trees (eg oak trees drop limbs in sustained hot dry weather.) As for disease or other causes, an experienced hunter in a familiar area could tell at a glance.
The message I got from the story is that they probably did have a very good reason. They either thought it would be too hard to communicate, or they were themselves cargo-culting the falling tree excuse when the reality was more likely to be... I dunno, snakes or nasty bugs or annoying sticky sap or whatever.
Like these guys probably have homes, with bedding of some sort, maybe they'd rather sleep next to their wives than some caterpillars. If I was giving somebody from a far-off place a tour of my workplace and, on the bus ride home, they suggested it was getting dark and we should camp on the sidewalk I'd probably not go for it. If they were really insistent I'd probably amplify the danger of sleeping on the sidewalk to shut them up.
The logic isn't applicable to any set of risks. As deadly as cars are, the risk of car crash death is much, much lower than 1/5000 per trip. It's probably closer to applicable to being a drunk driver, and "you would never operate a car drunk" is pretty accurate for many people.