Your thoughts vis-à-vis the Puritans led me to consider that it might have biblical origins pursuant to say, the devil, known as a serpent. Serpents have a forked or divided tongue and it is known to flick in and out very quickly so it is possible that they associated this action with bad or evil deeds. Something done lickety split might be considered fast and sinfully so.
Here’s a link to the column(by Dex) in question
I wrote that a long time ago, so I don’t really feel like I can comment. I guess it’s an interesting thought, but I certainly didn’t see it in any of the sources I checked. Like with so much in etymology, there’s lots of coincidences and lots of strange actual connections. So… ::: shrug :::: I got no clue. Anyone else? (We’ve got several people here who are very much into studying such things who might want to respond?)