But not all people in the audience would react like that to michaelkeenan's example warning. Some people would presumably value being informed of authoritative sources contradicting a claim that vaccines cause autism.

(And if your objection went through for fact checking framed as contradiction reporting, why wouldn't it go through for fact checking framed as fact checking? My mental model of an anti-vaxxer has them responding as negatively to being baldly contradicted as to being informed, "The Lancet says this is wrong".)