Much in line with my expectations, these are the figures I got from Google Books...
"if it ain't broke [don't fix it]" 223,000 hits
"if it ain't written [down, it didn't happen]" 183 hits (plus 45 more for ain't in writing)
It's also worth mentioning that there seem to be more different ways of following the (highlighted) search term in those 183 results than in all the 223,000 above it. Also that the "broke" version goes back to at least the 1890s, whereas the "written" version seems to have first appeared in the 1970s.
Taking that into account, I would say the "reason" for the "ungrammaticality" in OP's version is mainly because that's how the original was framed. And since that original had already lasted for the best part of a century, the latter-day aphorist was simply taking the advice of his chosen model.
I will admit I don't recall ever hearing OP's specific follow-on (it don't exist). Mostly I know the version I put above as a "general observation", and the unadorned "If it ain't in writing, it just ain't" when warning people about the (lack of) value in verbal agreements. But obviously the rationale for ungrammatically using don't exist in OP's case is even more that it's just an echo of the original.