He ate what had gone bad of the chicken.
This might pass grammatically, but it doesn't make sense. And not simply because nobody would eat chicken that had gone bad (they might, if they were desperate).
When we talk about "what's left of" something, as in your original example, it is a clearly distinguishable part of something. Some of the chicken had previously been eaten, and everything else was what was left of it.
Saying "he ate what had gone bad of the chicken" implies that he was able to distinguish between the good parts and the bad parts and ate only the bad parts. I don't think that is possible or logical. So perhaps it is just a bad example you came up with.
Comparable examples that make sense would include:
- He saw what was visible of the night sky
- She spoke to those of the class who had attended
- They did what was possible of the tasks