Pretty sure you can go into the settings and make it so the chat log doesn't show anything dealing with player chats (Yell, Shout, Say, etc) and then it will only show NPC/Event Dialog and Battle Dialog.
While not getting rid of it, it will still let you see all the NPC stuff and not have to deal with chat you don't want to see. Then you can just put them back in the log at a later time when you want.
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Can't sleep, clown will eat me.