The journal Nature Medicine published a major study about a cohort of over 105,000 people followed for 30 years. This is that researchers found.
Correlation isn't causation. But that's still interesting.
Very big caveat: while our primate cousins manage to eat some meat from time to time, it was our domestication of fire more than a million year ago that allowed us to access plenty of calories and nutrients from almost any food, everything was suddenly on the table, and was made easy to chew and digest. We are more omnivorous than the naturally omnivorous animals.