I am skeptical. Even if the numbers are lower, I can't believe no Amish children get cancer. Likewise, I think Type I diabetes can happen regardless of diet or other environmental factors. I do believe autism might be almost completely unseen. I think this is due to over stimulation from the modern world as much as from vaccines or other environmental factors. I bet some Amish guys can still be spergy, but are less likely to be all out autistic.
While it like all dietary info, it should be taken with a grain of salt since there's often an agenda, but in Gary Taubes book "Good Calories Bad Calories" he talks about so called "Diseases of civilization" where it wasn't so much # of calories eaten, but the types of calories, and overall life style, intermittent fasting common in societies where you had insecure food, where processed sugar was almost non-existent, and more physical activity was common.
Diseases of civilization were things like cancer, diabetes, heart disease etc.
Curiously, groups like the masaii of Africa, or Inuit in the far north who eat way more than their share of animal products and saturated fats, had far lower rates of those diseases. Additionally, during WW2 in England, when food and gas rationing were in place, they too enjoyed temporary reprieve from those issues, since they were eating less and moving more.
A life full of manual labour, natural foods, and plenty of fresh air almost seems impossible not to have many benefits.