I don't really have a clue, but I assume it goes way back. I would be surprised if it wasn't included when formal training for missionaries started, and likely before formal training.
That was taught IN Sunday School TO members, for as long as I remember. I believe it was built in to the missionary lessons for as long.
Without that scripture, how else could people test the spirit?
If you felt it burning, you knew it was true. And of course, as believers we felt it was right - so no need to verify or discuss the opposition. Which was straight from the adversary, so we were admonished to not give it a peep or a nod.
Sometime in the mid to late 60's there was a big push for every member to have a BoM. Until then, I don't recall anyone owning one. Nobody ever took books to church.
Once that happened, that scripture became one of the better known ones. Until then, most people didn't know or care what was in the BoM. My TBM grandparents never owned a BoM for the 90 years of life.