Ask someone why we celebrate Chanukah, and of course the first answer out would be about the miracle of the oil lasting eight days. This allowed the reconsecration of the Beis haMiqdash to be done at its halachic best, without relying on leniencies like “tum’ah huterah betzibur — tum’ah is permitted in public”. When everyone is tamei, no one is tamei. However, the Chashmonaim wanted to do it right, and therefore relied on the one tahor jug of oil for the eight days it took to produce more.
The earliest discussion of the laws of Chanukah is an appendix to Megillas Taanis, a list of dates from the late Bayis Sheini era which were minor holidays upon which declaring a fast was forbidden. I like the idea found in the Chida, the Eishel Avraham’s intro to Megillas Taanis and the Gra as for why there is no mishnah addressing the laws of Chanukah. (Although it is assumed and comes up in a number of places, so we know Rebbe considered Chanukah a holiday [Bikurim 1:6, RH 1:3, Ta'anis 2:10, MQ 3:9] with a specific Torah reading [Megillah 3:4,6] in which enough people lit something near their doorway that the person whose merchandise got burned by a Chanukah menorah is considered personally negligent and can’t sue for reimbersement [BQ 6:6].) They say that because it was already well documented in the appendix to Megillas Taanis, there was no need for a mishnah; and as you note — without need, there is no permissibility either. (Although why we assume this rule applies to rabbinic law rather than only interpretations of the original Oral Torah is beyond me. Also, the Gra’s son says his father speaks of “Mesechtes Chanukah” which I am only assuming is the appendix.)
But there is no mention there of the miracle of oil in Megillas Taanis. Nor in the Al haNissim we insert into Shemoneh Esrei and benching. In the Apocrypha, the reasons given relate to winning the battle for the Temple Mount, and the subsequent celebration of a quasi-Sukkos for eight days to compensate for the missed opportunity to celebrate Sukkos at the Beis haMiqdash while it was in desecration. The latter explains Beis Shammai’s position, that we light 8 lights the first night, then seven, then six, etc… to parallel the cows of the Musaf offering of Sukkos, which also decrease over time: 13 the first day, 12 the second, and so on. Clearly in their time, the connection to Sukkos was still a given.
The miracle of the oil would also be an odd reason for a holiday. How many people could have witnessed the miracle? The subset of kohanim who were tahor and working in the Heikhal that week so frequently that they can attest that no one refilled and re-lit the menorah while they were elsewhere. But a major feature of the importance Judaism ascribes miracles is their public nature. The entire word neis, miracle, is more literally a flag or military standard, something that calls attention to Hashem’s Presence. We have no (other?) holidays set up to commemorate private miracles. I therefore think it makes sense to take the book of Maccabees and Al haNisim at face value and say the holiday was at that time about the restoration of some level of political autonomy and of Temple worship for the next two centuries.
The first mention of the miracle of the oil is in the gemara, written centuries later. The gemara goes off on a tangent in the middle of the laws of Shabbos lights to discuss those of Chanukah. At some point (Shabbos 21a) it asks, “Mai Chanukah — What is Chanukah?” and answers with the miracle of the oil. But given that we know that Chanukah was codified even before the mishnah, anchored — even if in a few mentions — in the mishnah, how could the rabbis of the talmud not know what Chanukah is about? And why is the answer one that was not given in any of the texts we have from before the gemara?
So I would assume this gemara records a conscious attempt to change the theme of the holiday. When instituted, Chanukah was about the restoration of the Beis haMiqdash and the autonomy possible under the Hasmonean kings. But then we lost it all. No autonomy, the majority of the community of the land of Israel forced to join their brothers in exile, no Temple.Notice it’s the Babylonian Talmud that is asking this question! The laws are on the books and they weren’t empowered to repeal a law enacted by a Sanhedrin in the Lishkas haGazis in the Temple. But the meaning was gone; rather than being a celebration, it became a reminder of everything lost.
And so the amora’im set out to reassign meaning to the mitzvos of the holiday by emphasizing a miracle than until then was a tangential thing — but at least related to the central mitzvah of the holiday, lighting the menorah. The Talmud isn’t asking “What is Chanukah?” in the abstract theoretical plane, it is asking pragmatically. In a time of exile, when Hashem’s influence in world events is more hidden, the most inspiring part of Chanukah is the one small way Hashem showed that the victory wasn’t merely incredible milirary prowess and good luck, but His intervention, the one explicit violation of the laws of nature — the miracle of the oil. Chanukah is very much a festival of light, reinvented as such in the darkness of exile.