I get the same result – I am assuming it is a bug. Putting in a new citation after the no author, before the repeated citation restores the author to the 4th citation… Very strange.
But – If I change the footnote Author list setting from "omit the author list (for subsequent works by the same author) it is fixed. I am not sure what that might break though. (I unticked the box)
Note 2 – the setting for footnote templates has a box ticked “When using the short form, include the title field only when needed to disambiguate a citation”. If you untick that box and save as and then change the output style to the newly modified one, at least the title shows up, but still no author for the 3rd repeat citation…
Anyway, a copy of the style with these two changes is attached.
Chicago 16th Footnote (disambiguate change).ens (80.2 KB)