I have seen questions and discussions which argue whether the miracle of Channukah was the military victory (rabim b'yad me'atim) or the finding/lasting of the pure oil. However, it seems that neither of these might be the case.
According to the Chabad page, "the purpose of the Chanukah lights is simply to remind us of the Chanukah miracle." The candles are not an literal reminder of oil, but a reminder of the miraculous events of the story (or the Channukat Hamizbe'ach, according to the Chiddushei Aggadot on Shabbat 21b)
In the Kitzur Shulchan Aruch (193) we read that women have the tradition of not working while the candles are lit (or 30 minutes, so no exploiting those 24 hour candles...). This expression of respect for the candles is mentioned here, citing the Mateh Moshe. It is further expressed on that page and this one that a woman's connection relates to the "first nights" issue and the daughter of the Kohen Gadol's (and/or Judith) bravery in killing a lascivious Greek tyrant (mentioned in the Taz to S"A 670:2:10). The Mishna Berurah to 670:1 writes,
לפי שנעשה נס על ידיהם because a miracle happened by their hands
If the actions of this woman caused the invaders to flee (as referenced in the Kaf Hachayim, 670:2 "וכראות החיל כי מת גבורם וינוסו" and even moreso, the mishna Berurah "וברחו כולם"), and this action is the miracle that inspires women, (and some men according to the Maharil, as referenced in the Kaf Hachaim, 670:1), to not do any melacha while the candles which remind us of that miracle are burning, then why isn't this the true miracle of Channukah?
This seems to be more explicitly stated in הערות הגרי"ש אלישיב מסכת קידושין דף לד עמוד ב as
עיקר הנס היה על ידן, בפורים ע"י אסתר בחנוכה ע"י יהודית בפסח שבזכות צדקניות שבאותו הדור נגאלו, והדרנא לדינא כרע"א. (ועי' קוב"ש ח"ב סי' ל')
[side note -- the Lev Meivin differentiates in an understanding of "af hen hayu b'oto haneis" between a miracle happening through women or women being subjected to the evil decrees in general]
There needs be no "v'rabim b'yad m'atim" military victory nor any (possibly later) concept of oil. The Channukah events (though is it a miracle that cheese makes one thirsty, or wine, sleepy?) led to the victory and rededication of the temple (as the Chabad site writes, "the miracle itself came about through the heroism of a woman").
So why is this not the "miracle of Channukah" that would be mentioned explicitly in prayer and song?