Moderation on the main site is pretty silent and unobtrusive. If you were not involved in a situation, you probably won't notice that anything is happening unless you visit the involved posts by chance. Certain kinds of misbehaviour are far noisier, but most situations can be handled without drawing too much attention.
Chat moderation is the complete opposite of that. A spam/offensive flag notifies every single 10k+ and chat moderator that is online at that moment. There are around 25-50 moderators logged into chat at all times, and the number of 10k chat users is likely even higher.
The result is that if flags are starting to be cast, you get a whole lot more users involved than you actually need to deal with the issue. A chat flag is a big huge blinking sign saying "Hey, here is some drama happening in chat. Get the popcorn and lets watch the trainwreck". While watching such drama can be occasionally entertaining, as long as you're not actually trying to do something about it, this does not really help deescalate the situation.
Too many users getting involved often just makes it worse, and it also leads to endless discussions on what exactly went wrong this time, who's fault it is, why the mods are abusing their powers and why the chat flag system sucks.
Chat moderation is harder than moderating on a main site in my experience, one of the main reasons is the realtime nature of chat. But the flag system and all the attention it draws to every minor issue is not really helping.
How could the chat moderation system be changed to address this issue? The current system was designed for very fast response times, which can be necessary in a realtime medium. How could we still keep the fast response times but avoid broadcasting the drama across the entire SE chat?