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?
I know this question is many years old and I'm a bit late, but I was surprised that no one quoted the footnotes to the ArtScroll Siddur, which clearly give reasoning to this.
In many congregations it is customary to divide the Sabbath and Festival services among several chazzanim: one for Pesukei D'zimrah; another for Shacharis; and a third for Mussaf. On the Sabbath, the chazzan changes at שׁוֹכֵן עַד, He Who abides forever, because the Sabbath was the climax of creation, when God had all of creation, including man, to acknowledge and praise Him; on Festivals the chazzan of Shacharis begins הָאֵל בְּתַעֲצֻמוֹת, O God, in the Omnipotence, because the Festivals testify to the Exodus, when God revealed 'the omnipotence of His strength;' however, on Rosh HaShanah and Yom Kippur, when God is 'King sitting in judgment', the chazzan begins הַמֶלֶךְ, the King (Levush).
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.
The Leader begins at different points on different holy days of the year. On Shabbat he begins with "He inhabits eternity," emphasizing creation; on Yom Tov, with "God - in Your absolute power," laying stress on God as He acts in history; on Rosh HaShana and Yom Kippur, with "The King - enthroned," evoking ideas of justice and judgement.