HaLeiVi – ” We thank Hashem for the Yeshua as well, but Chanuka was not established because of it.”
Indeed, in T”R “my chanukak” (Rashi – al eza ness kvauha – for which ness was chanukah established) chazal answer the question by elaborating about the ness hashemen, making THAT the ikar ness.
However, in the Al Hanissim, there is hardly (if any) mention of the ness hashemen, rather ONLY the ness hamilchoma, making THAT the ikar ness!
The Meiri on Shabbos holds that the first day’s lighting of the menorah was to commemorate the nitzochon hamilchoma. (Which BTW is an answer to the B.Y.’s kasha).
As such, the pisuma nisa of the first day’s lighting covers the ness hamilchoma too.