As I understand it, the liver issues mostly came from unscrupulous extract sellers who were using the stems and leaves.

Basically, you drink the dried & powdered roots of the plant, but for a while there was a big boom on making pills with extracts. Demand exceeded supply and some people started using the parts of the plant not traditionally consumed. These parts have some other compounds in them that you don't want (similar to how tomato and potato leaves contain solanine), and now people were taking basically concentrates of them. This is where you get problems. The roots have hundreds of years of traditional consumption in Polynesia, the leaves, not so much. Supposedly they were traditionally used as an abortifacient, so probably not something you want to drink.

Among those who drink traditional varieties of kava in Polynesian cultures, I do not think the liver problems have been well documented. Which doesn't mean it doesn't cause problems, it could also be due to it being a relatively less studied topic.