I can follow much of what you have written in the previous post, however you seem to ignore the fact that the German Kingship even ITTL developed out of Frankish kingship (East Francia).
Not entierly so : around the time of Henry of Saxe's election, the kingship was said to be on "Teutons", and no longer on Franks which were identified with Franconia.
With the establishment of national dynasties (there Franco-Saxon), identification with Frankish kingship (especially when this one survived into a foreign form in Francia) was less obvious. It survived IOTL because of the reclamation of the imperial title that in Latin Christiendom could only be traced back to Charlemagne.
I assure you I'm not ignoring that Germanic kingship was issued from Frankish, but I just want to point that in a HRE-less Germany, it may not be seen completly as such with the absence of an imperial title and the existence of a Frankish kingship beyond western borders.
Eventually, claiming the imperial title made a lot, IMO, to strengthen this bond : it was virtually absent in France before the XIVth century while it was the most direct continuity of Frankish kingship.
Henry the Fowler, duke of Saxony (or the Saxons) was elected king of East Francia.
Actually, they elected him "Rex" IRRC, without more precision. (As Otto I was, probably crowned during his father's reign).
East Francia (
Orientalis Francia) cease to be used for the whole kingdom approximatly in the late IXth century, while
Teutonicum regnum is used (explicitly in replacement) in the early Xth century by Germans.
That said it's still used at this date and later, but for Franconia only (in order to distinguish France from Franconia).
Admittedly the process took time, and you can find
Francia Teutonica and
Francia Gallica but these refers to the peoples rather than the kingship.