but it still doesn't explain how the term gadżki (and gacie and variants) came into common use
The word "gadżki" does not exist. You might mean "gatki" - an abbreviation of "gacie".
The word "gacie" is and was a very common word. It didn't have to "come into use" - it WAS in everyday use, and still is as regional / slang word in Polish, and a regular / dialectal word in many other Slavonic languages.
As per your earlier post:
"You've certainly made some interesting points, however they don't quite ring true"
" and seem like you're trying to prove a pint."
Well, yes - I am trying to prove my point. What's wrong about that?
"One that isn't backed up by any evidence".
Except the history of Slavonic languages, that is.
" There is however a traceable history of the word's use in Yiddish."
I am not saying the word was not used in Yiddish. It obviously was / is. But it seems to be a borrowing from the Slavonic languages, not the other way round. Or is "kasza" also a borrowing from Yiddish, as "kasha" is a legitimate Yiddish word? How about "boychik" - is "boy" an English borrowing from Yiddish?