Linking this anecdote to anti-vaxxing really seems a stretch, and I would like to see the reasoning behind that. My impression is that anti-vaxxers have more issues with vaccines themselves than with doctors who recommend them
I think that completely misreads a comment that was already painstakingly clear, they're specifically talking about the phenomenon of reasoning by anecdote. It wasn't a one-to-one equivalence between LLM driven medicine consultations and the full range of dynamics found in the anti-vax movement. Remember to engage in charitable interpretation.
They are closely related. The authority of the medical establishment is more and more questioned. And whenever it is correctly questioned, they lose a bit of their authority. It is only their authority that gets people vaccinated.
