trend of anti-intellectualism and distrust of doctors
As someone who knows a practicing doctor who is also anti-vax, these are orthogonal issues. Sometimes, distrusting a specific doctor is the more intellectual approach.
You'd be surprised how many physicians and nurses refuse vaccination. You'll just never hear about it. AMA is one powerful beast, I wish I had a union like that.
However, sometimes fun little things like this happen that show their true colors:
"Starting in early 2003, the United States government started a program to vaccinate 500,000 volunteer health care professionals throughout the country. Recipients were healthcare workers who would be first-line responders in the event of a bioterrorist attack. Many healthcare workers refused, worried about vaccine side effects, and healthcare systems refused to participate. Fewer than 40,000 actually received the vaccine.[29]"
That doesn't mean distrust of vaccination in general - at least by doctors. It was for a potential bioterrorist attack. It perhaps more reflects the low likelihood or belief that there would be a small pox attack.
If you assume the low likelyhood of the attack was the reason, that means 90% of those physicians could be lying. The stated reason for refusal was concerns about side effects. It's in the quote.
What you stated as a fact, 96% vaccination rate, is actually a self-reported survey.
Why do a survey when public health CDC records could simply be matched with the physician licensing registrars?
Seems an automatic search like that would save physicians their valuable time, aren't they very busy with a pandemic right now? Instead of hard data from CDC, we get self-reported, likely anonymous, self-reported survey.
It's always cost-benefit. Relative risk of side effect directly relates to likelihood.
Risk of side effects vs benefit of vaccine.
I am not likely to take an HIV vaccine, since my personal chance of contracting HIV is incredibly low. So any side effect isn't "worth it" -- even a sore arm. But that doesn't mean I'm anti-vaccine.
I also don't wear a bullet proof vest around because it's too heavy ("side effect"). Does that mean I'm anti-bullet proof vest? No. But I would wear a bullet proof vest in a war zone -- even if it's heavy.
If there was a widespread small pox outbreak in the U.S., I'm certain more than 10% of physicians would take the vaccine. Does that mean they were lying before? No.
Because even if I did point to official stats, you would say they are just lying?
A few months prior to the AMA survey, Long Term Care Facilities reported a 75% vaccination rate amongst physicians at their facilities. So presumably higher now.
The stated reason for refusal was concerns about side effects.
As another comment already mentioned, smallpox vaccines (at least historically) tended to have undesirable side effects, like permanent scars. Smallpox vaccination is probably not a good proxy for vaccination overall. It's not worth getting vaccinated for smallpox unless you expect a decent risk of exposure.
As someone who knows a practicing doctor who is also anti-vax, these are orthogonal issues. Sometimes, distrusting a specific doctor is the more intellectual approach.