You'd be surprised how few people are interested in evidence these days. More often than not, your Reuters/AP source will be shoved aside in favor of some page along the lines of reallycoolnews.blogspot.com.

We've already seen this happen–when YouTube started putting labels on all COVID-19-related content, deniers were unsatisfied by the sources presented because they have been indoctrinated to believe nothing mainstream can be trusted.

Most people don't actually believe things or care if they're consistent (and rightly so.) They just say stuff to hear themselves talk.

The correct answer is to prevent them from saying things.

It'd be like trying to talk to a 9/11-truther or a flat-earther, they'll have a refutation ready for all of your evidence, or they'll convince themselves your cited sources are part of the conspiracy..

They'll use any excuse to delude themselves that they're correct, because to admit otherwise is to admit that they were dumb enough to have been conned by a terrible conman. When they saw the morons attacking congress, they convinced themselves "people wearing helmets are Antifa, and all the attackers are Antifa plants...".