There is no longer any evidence that eating fatty foods causes obesity.
Unfortunately this myth still hasn't died, though it's probably going to in
a few more years.

As for people 80, 90, 100+ years old: The Georgia Centenarian study showed
that most people who live to be 100 or more tend to ignore medical advice
on healthy diet and do, in fact, tend to eat fatty foods, meats, and lots
of other supposedly "unhealthy" foods. Almost none are vegetarians, almost
none restrict calories, and all of them tend to avoid calorie-ristricted
diets.

As for humans and meat: the average intake of all hunter/gatherers
worldwide shows that on average they get slightly more than half their
daily calories from meats.  The African !Kung are on one end of the
spectrum with only about a third of their intake as animals, however, they
live near an enormous mongogo nut forest and so get most of their calories
from that, with meat making up only about a third of the total diet.  On
the other end of the spectrum are the Inuit, who eat a nearly 100% meat
diet.

Nowhere among hunter/gatherers are there any vegetarians.  Nor is there any
evidence that anywhere in human evolution was there ever any vegetarian.
Humans have always eaten meat--always, since long before we were human.
The very earliest human ancestor (australopithicus) ate meat, and if you go
back to our pre-human animal ancestors, you're basically looking at small
omnivorous animals who ate insects and animal eggs amongst other things.

There can be no support whatsoever for vegetarianism as a "natural" diet
for humans.  The foods you eat as a vegetarian may be "natural" in that
they are organically grown or whatever, but as far as being natural to
humans, it isn't.  At all.  And there's a good bit of evidence that
vegetarianism is, long-term, a potent health hazard to many people,
although there is also some indication that short-term excursions of a few
weeks or months into vegetarianism may be healthy for some individuals.