I had sent a few Yekkes that I knew, a publication about the customs in Frankfurt. It is comprehensive and revised yearly. It was sent before Rosh Hashono. I noticed though that it stated that the Kittel was not worn by the Shaliach Tzbibbur on Shmini Atzeres.
I do not know if Hamburg had a different Minhag, nor what Berlin let alone Poland/Russia followed, but [hat tip RDS] the following seems to also confirm (unless I have misread) that it was something that was generally worn originally on Yom Tov, over the clothes, as a special Malbush Yom Tov (which is a Halacha). [Those of you with access to one of those white cylinder style hats can wear those 🙂 I can remember the days when there were heaps of them adorning the heads of Mispallelim … they used to get yellow over time, as I noticed] I couldn’t find a picture of one but they are like a flatter floppier version of the Chazan style tall hat (not to be confused with the Shloof Yarmulke, which is another story)