I always thought the term "bug" came from people finding a literal bug stuck in the machinery of an early computer [1], TIL that Edison was using the term in the 1800s.
[1] - https://education.nationalgeographic.org/resource/worlds-fir...
So no.
The earliest written evidence of the word "bug" used in this context is from the 19th century. By that time, most machines were made from metallic parts.
Of course, more primitive machines were still using wooden parts, as they do today! Of course, the word "bug" might have been used in this context centuries earlier, without producing any written records!
But let's not kid ourselves. OP had no evidence of their theory whatsoever. They just made up a story.
I can make up an equally "likely" theory: maybe people were often saying "oh, I've made a big mistake", and "big mistake" turned into "bug mistake", which was then shortened to "bug".
No surprise that the word bug itself is etymologically linked to goblins or ghosts in the time period referred to. And fits perfectly with the usage in the quoted paragraph.
The probing question is whether such failure modes (from bugs) were indeed common in pre-industrial machines (so as to warrant popular expressions). Thats something an actual expert could opine on with some impact.
It needs to be called out, before people start misquoting it as fact, like how people ran with the 100% pseudoscientific glove-knitting theory for Roman dodecahedrons, which some people still quote on this board as if it had any scientific merit at all.