Why does the Shaliach Tzibbur start Shacharis on Shabbos from Shochein Ad, on Yom Tov from HaKeil, and on Rosh HaShana/Yom Kippur from HaMelech?
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Levush (Orach Chaim 488:1) says that we start with הא-ל בתעצומות on Yom Tov, because all of them are "in remembrance of the Exodus from Egypt," when Hashem displayed His mighty power. He also says (ibid. 584:1) that we start with המלך on Rosh Hashanah and Yom Kippur (and change the wording to המלך יושב, "the King is sitting"), because these are the times when He is sitting on His throne of judgement.
Not sure about Shochein Ad on Shabbos.