Comparing models with search tools to models without - when there's no option for "I am unable to answer this question without access to search" - doesn't make sense to me.

Yes, so in that case you set them up to disagree and then measured disagreement.

The title mention "fact-checks", but "fact checking" is a process in which facts are checked against sources, not one where you are given a random fact and have to tell if it's true or false from your own memory. That's what is normally called a quiz game. So a more honest title for this research would be "Models answer differently to quiz questions".