I think you are a bit confused because you mixed up negative and affirmative forms.
I was mixing up two things, actually -- (1) genitive after negation and (2) partitive genitive after verbs expressing consumption (съесть, выпить) or nouns expressing quantity (like стакан, горсточка, ложка).

I mean, wouldn't it be possible to say "Она добавила горсточку сахару", "She added a small handful of sugar"? Or would сахара be preferred there?

And I'm pretty sure that "я выпил чая" and "я выпил чаю" are both okay (with a partitive meaning that's different from "я выпил чай").