Well I didn't actually get them banned, but since I haven't seen them on Twitch for nearly a year now they might be gone anyway.
Strangely enough another person who I often saw in another streamer's chat, got banned recently, not suire what for, but it must have been bad. I didn't have a problem with that person though.
The problem is that the block button doesn't stop the person from talking about you. They can still insult you, spread rumors about you and destroy your reputation. Personally I think the block button makes things worse. It stops you from seeing the person's attempts to attack you which prevents you from shooting their BS down.