We’d love to hear your pain points in this regard. 
If we can address the “ease” of moderating chat, everyone would be happier! 
I got myself a “timeout” from here on Meta yesterday, to experience the moderation from the perspective of both an admin and a trouble-user.
Flagged messages from chat came up in the review queue, and I was able to take similar actions on the messages, including deleting it from the queue page. 
Then I was unable to interact on the site for a while (though I could still read), the same user flow for moderation of topics.
Of course this doesn’t address the other side of moderation, which is who is flagging/reporting. I think I run into this less often because I don’t participate in “firehose” chat channels, so objectable content doesn’t “fly by”, we have managable feedback loops for our various community sizes.
Exploring this made me realize: a lot of different configurations of people use chat.