The claim is that a doctor over 100 years ago named Weston Price published a book based on data he collected around 1900 or so that linked poor dental development to the Western diet and good teeth health to a "traditional" diet, generally high in animal foods.
Price doesn't offer any evidence that his theory is backed by anything more than a mixing up of correlation and causation.
For the vegan diet, Price was looking at unfortified foods, limited to what could only be grown locally and that lacked many of the good characteristics of modern vegan foods. And as far as the omnivorous diet, he was looking at locally sourced antibiotic-free, hormone-free, non-factory-farmed animal foods, a vanishingly small fraction of what modern Americans eat. In short, this is looking at the best animal-derived food that almost no carnist eats, and the oldest, weakest unfortified vegan foods that, again, almost no vegan eats. So the relationship between these two things has nothing to do with anything in reality.
The idea that veganism is reliant on processed foods as carnists citing Price's work would define it is just not true or poorly defined
Is really the best epidemiological data on veganism from the 1930s? Surely, if vegan diets were not safe it would be the scientific consensus based on modern data, not some theory that Grandpa cooked up in the days when dying with a rag on your forehead was cutting-edge medicine.