I would have thought that catching someone’s drift was a peculiarly modern idiomatic way to understand them but like many “modern” things, this one may have begun with our man, Shakespeare. I don’t have any evidence for this but is it not impossible that Shakespeare kicked this particular idea off?
Someone’s “drift “ caught on
While a “Fetch” did not – at least not in this context.
It does make SENSE though – the little lie acting like a stick in a game of fetch sending a dog chasing after something if only in a game.