The Shu"t Shalmey Chova (#107) discusses can one say the ברכת הרואה on the lighting of a katan. He assumes the question is whether the action of the kid is considered a מעשה מצוה. He doesn't note the Butchatch (679) already raises the question. (It is interesting the Butchatch notes his missing the time to light the candles on Erev Shabbos.) Rav Tzvi Ryzman wants to say a different approach to the question, the question is not if the katan's lighting is a mitzvah lighting, but the question hinges on if the ברכת הרואה of שעשה ניסים is a birchas mitzvah of birchas hashevach akin to saying a beracha on a place that a miracle happened. If it's the first way, a birchas hamitzvah, there has to be an actual lighting of a mitzvah to say the beracha, which is not the case when a katan lights, but if it is a birchas hashevach, it would be possible to say the beracha even on the lighting of a katan. His explanation seems problematic for even if the geder of the beracha is a birchas hashevach, it needs to be recited over a חפצא של מצוה, would anyone contemplate saying a beracha on the lighting of a goy?
As mentioned by my father in the past, the Rambam's opinion is that the lighting of the menorah itself is a fulfillment of hodah, not a separate mitzvah.
The Gemarah Megillah (14a) says ת"ר ארבעים ושמונה נביאים ושבע נביאות נתנבאו להם לישראל ולא פחתו ולא הותירו על מה שכתוב בתורה חוץ ממקרא מגילה מאי דרוש אמר רבי חייא בר אבין אמר רבי יהושע בן קרחה ומה מעבדות לחירות אמרי' שירה ממיתה לחיים לא כל שכן. Rashi asks but ner Chanukah was added? He says that wasn't an addition of prophets for there were no prophets any more. The Ritva asks but why did they add ner Chanukah? He explains it is the same reason as to why they added Megillah. According to the Ritva just as the Gemarah considers Megillah a form of shirah, ומה מעבדות לחירות אמרי' שירה ממיתה לחיים לא כל שכן, so too ner Chanukah. In other words the lighting is a form of shira. It is not just an act of a mitzvah.
The Rambam Chanukah (3:3-4) says וְהַדְלָקַת הַנֵּרוֹת בָּהֶן מִצְוָה מִדִּבְרֵי סוֹפְרִים, כִּקְרִיאַת הַמְּגִלָּה. כָּל שֶׁחַיָּב בִּקְרִיאַת הַמְּגִלָּה, חַיָּב בְּהַדְלָקַת נֵר חֲנֻכָּה. Why does the Rambam compare Chanukah to Megillah? In light of the Ritva, it is understood. The Rambam is explaining that in the same way Megillah is a mitzvah as a means of expressing shira, hoda to Hashem, so too ner Chanukah. The Rambam continues that all who are obligated in Megillah are obligated in ner Chanukah. Why does he connect it to expressing who the obligation is upon? Woman are obligated in these two mitzvot because אף הן היו באותו הנס. Tosfos asks why we don't employ the same logic to tell us women are obligated in sukkah, or for that reason, they should be obligated in the four cups of wine? The Rambam is telling us that אף הן היו באותו הנס doesn't create an obligation when the miracle is the mere reason for the mitzvah; it is only when the miracle itself defines the mitzvah as a cheftzah of hodah that אף הן היו באותו הנס creates an obligation on a woman (Rav Solevetchik in Mesorah Journal volume 13).
Tosfos Shabbat (21b) asks how the seal on the bottle of oil helps; it should still have been tameh since there is a gezerah that goyim are tameh like a zav. The movement of the jug would have made it tameh? There are a few answers to the question. The Bach (670) suggests that it had a small opening that one could not fit their finger into, and something that cannot be tameh through touching when in the vessel does not acquire tumah through movement either (Shabbat 84b). He adds that it still had to be sealed; otherwise, we would need to be concerned that maybe the oil had been removed and replaced, or it had become tameh through tumas ohel. Therefore, there must have been a צמיד פתיל on it. R. Akiva Eger asks that this contradicts an open Gemara in Chagigah (25a) that צמיד פתיל does not work for hekdesh. In שו"ת קרית חנה דוד he suggests answers based upon the assumption that the din that צמיד פתיל doesn't work by hekdesh is only Rabbinic, and in this case, they did not impose the gezerah. As discussed in an article about preparing oil nowadays, there is a machlokes Rishonim about what level of kedusha is necessary for the law of צמיד פתיל not to apply. The Rashash in מדייק from the Rosh on Kelim (10:2) that only on kedushas haguf the צמיד פתיל doesn't help, but it does say something that only has קדושת פה. Tosfos Temurah (14a) says the oil for the menorah did not have kedushas haguf. The Chazon Eish (Menachot 30:7) says it is not logical that the oil never obtained kedusha at all. Tosfos means it didn't have kedushas haguf, but did not have kedushas peh. Putting that together, the oil for the menorah would be eligible to be saved by a צמיד פתיל (From here.)