“as in not having the authentic tav dagush”
Don’t you mean “tav rafeh”? A lisp can be described as pronouncing a shin as a tav rafeh.
Regarding your other speculation, it is a well documented scientific fact that the nucleus of galus ashkenaz came from bavel. The division between the rishonim and acharonim was when — to put it poetically — the kesser Torah moved to Ashkenaz.
That was why ashkenazim lost the vav (pronounced whoa) and gimmel rafeh, pronounced (ghhhhh); there is no “w” or glottal (I think) “g”in German.