Q. Why in some shuls the chazan begins singing Hamelech from his seat and then approaches the amud?
A. See similar question 1776 where we answered:
“Horav Shlomo Miller’s Shlit’a opinion is that on the Yomim Noroim we are requested to be “machriz,” proclaim and assert that Hashem is king. Therefore, we do it in a more explicit and expressive way.
Horav Ribiat Shlit’a quoted from seforim that explain that the Talmud (Gittin 56b) relates that when Vespasian demanded from Rabi Yochanan ben Zakai; “If you insist that I am the king, why then did you not come until now?” Therefore, the shaliach tzibur proclaims that Hashem is king, as soon as he begins his service, even before he reaches the amud.”
Horav Dovid Bartfeld Shlit’a contributed two more reasons, namely: The Barditchever Rebbe compared it, to a coach being attacked by robbers and out of desperation one traveler screams aloud; The King! I see The King! And the robbers, out of fear run away. We too scream Hamelech from our seats, so that the Soton, that tends to rob us from having proper concentration and devotion in our tefilos on this most crucial and important day, should run away.
He also quoted Horav Melech Biderman Shlit’a that everyone experiences the presence of the King of the Universe in his own particular way, understanding and perception, so to say; from his own seat.