If you aren't opposed to Google, "Keep" seems to be fairly well suited for this kind of thing. You can share to it easily on android, it supports a bunch of organizational things (labels, colors, archiving) but they're not in the way if you want to ignore them, and the browser version works great. If you're attached to the chat style, then it might not be perfect, but it's easy to use it as a continually appended log style experience.
