There are two ways to interpret this "growing evidence". One approach, favored by this author, is to say their old wild diet was healthier and therefor those eating in such a fashion were less likely to get degenerative diseases.
The other approach is to say they died too young from injury, starvation, or infections. Had they lived longer they too might have contracted more of our degenerative diseases.
I strongly suspect lifestyle may have played more of a role than diet alone. The modern sedentary lifestyle with enormous quantities of extra calories at our fingertips is quite a recent invention of humanity.
Getting more nutrients back into our food is good. So too would be to increase the variety of foods we eat (monoculture seems to not be healthy for the environment or our bodies). But I suspect the biggest gains in well-being will come from rethinking how we apportion our days and incorporate more physical activity and leisure time into them.