As far as I understand, piyyutim (poetic additions to/replacements for certain brachot in public prayer) are optional/only a minhag, and that omitting all of them wouldn't be a problem.
On the other hand, Rabbi Hershel Schachter mentioned in a recent video (here, from 1:10) that the Seder HaAvodah (a piyyut describing the day's proceedings in the Temple — or rather, one of many different such piyyutim) in musaf of Yom Kippur is required, "midina degemara" (by Talmudic law). I know that the Seder HaAvodah was said during the time of the gemara (Yoma 36b, 56b; it wasn't in poetry then), but I don't see where it's described as required. Indeed, there are communities who don't say the Seder HaAvodah during the repetition, but instead say it afterward (although I know of no current community that omits it entirely, but ShimonS points out that Rambam and the old Aleppo rite omit it).
One possible source: Rav Amram Gaon describes the Seder HaAvodah here as חובת היום (requirement of the day), but I think he's actually talking about the sprinklings and viduy, which are the requirements of Yom Kippur in the temple, and not the telling of the Seder in the musaf repetition. In any case, I wouldn't call Rav Amram Gaon's requiring this piyyut as being midina degemara, so I'm interesting in other (earlier) sources.
Is it really required, whether midina degemara or otherwise? If it is, what is the status of a "musaf service" that omits the Seder HaAvodah (both from the musaf repetition and afterward)? If it isn't, is there a strong a priori preference for including it, or can one omit it intentionally?