Yom Kippur Musaf includes a piyut describing the avodah that the Kohen Gadol does on that day. Why is the whole thing said at Musaf? Wouldn't it make sense to say
- the part about the history of the world... at some point, maybe on Rosh Hashana;
- the part about 7 days before Yom Kippur 7 days before Yom Kippur;
- the part about erev Yom Kippur on erev Yom Kippur [the oath is a particularly nice connection to Kol Nidrei];
- the part about the terumas hadeshen at Maariv;
- the part about the Tamid at Shacharis;
- the next large chunk at Musaf;
- the part about the Tamid at Mincha;
- and finish it either immediately or at Neilah?
Obviously it wouldn't make too much sense to split אמיץ כח in this way. I'm not sure about the other piyutim that are used or have been used in the past. But that's just because of the composition; it would have been possible to compose a piyut with splitting in mind.
Why describe the whole avodah at Musaf, even the parts like the Temidim that by definition have to happen earlier and later?
And are (or were) there communities that split up the avodah similar to what I've described here?