I observed a shul where the Shaliach Tzibbur left the Omud after saying Korbonos and only returned to say the concluding part of Pesukei Dezimro. What is the origin of this practice?
The Tur (סימן נג) writes
ועומד ש"צ ומתחיל ישתבח
Translation - the Shaliach Tzibbur gets up and says Yishtabach.
We can derive from this, that until Yishtabach no Shalich Tzibur was present, as he only "gets up" now.
The מור וקציעה explicitly writes that the minhag in פולין ומעהרין is that the ש"ץ only approaches the amud by yishtabach. This minhag is also mentioned in
- כל בו
- שו"ע הרב
On the other hand -
When the Shulchan Aruch when brings this halacha uses different words
אומר שליח ציבור ישתבח מעומד
Translation - and the shaliach tzibbur has to say Yishyabach while standing.
The תורת חיים is מדייק that the reason for the mechaber's choice of words is to point to the minhag of the ש"ץ already being at the amud from beginning of the תפילה.
An additional דיוק is the mechaber's psak (51:3) that if one hears the חזן say the beracha of Baruch Sheamar one has to answer Amen. Obviously, there must be a chazan at the amud for this to ocur.
The Beer Hetev (48:1) also talks about the importance of saying Korbanos together with the ש"ץ. Another דיוק that the minhag was to have a chazan present from the beginning of davening.
As to the contemporary minhagim-
You will often find yeshivishe minyanim without a chazan until Yishtabach/שוכן עד, especially if there is no chiyuv. Chassidim and non-yeshivish ashkenaz minyanim will generally have a chazan at the amud from either the beginning of davening, or at least from kaddish before hodu/mizmor shir.
For further reference see -מנהג ישראל תורה, אורח חיים, סימן נ"ג