As someone who spends (spent?) a lot of time in chat, with a hand in moderating it - I was a fairly active moderator on the general Stack Exchange chat server.SE when I had my diamond, and I am also currently a room owner in the Tavern on the Meta on Meta Stack Exchange Chatchat.Meta and the Meta room on Stack Overflow Chatroom on chat.SO - I have some experience with how this system works in different contexts.
When there aren't any moderators around and the room has to rely on the room owners alone - like the Tavern for a large partmost of its history - the tools that the room owners have simply aren't powerful enough. Sure, you can keep kicking someone from one room, but there is no way to stop them from then going into another room, or creating their own, and causing trouble there, and no easy way to stop them from creating new rooms. A room owner's capabilities are limited to a single room, and they don't have any way of dealing with an issue that escapes that room.
When there are too many moderators, like the Teachers' Lounge, it also becomes impossible - since all users there have moderator access and there is no way to moderate the moderators, any problem will simply keep devolving since there's no way for anyone to step in forcefully (other than by an employee revoking moderator privileges on all sites).
Now, all of these problems, I believe, stem from one main issue: weWe expect the diamond moderators to moderate chat.
However, this doesn't work. The site moderators have been elected / appointed for their skill in moderating a Q&A site. The skills required for effectively moderating a Q&A site are, if not totally different, largely different from the skills needed to moderate a chat room / server. It's a different skill setskillset to moderate a live chat vs. a Q&A site. Some of the moderators on the sites have those skills. Others... don't.
These chat moderators would need to be people who are able to effectively moderate chat, not the main sites. Aside from the extra abilities to moderate the moderators, these chat moderators would also have all of the current capabilities that diamond moderators and room owners have, and to have them across all threeof chat servers (chat.SOSE, chat.MSEMeta, and chat.SE)SO.
(This might mean that per-site moderation rooms - such as those on MSE or SO - would need to be adjusted to only allow the site moderators in. Chat moderators should not have access to the site-specific private rooms. Whether or not they should have access to other site-hosted private rooms (as opposed to de-parented rooms like the TL and Good Subjective) is something I'm not sure about.)
This way, the hierarchy of chat moderation would go: regular users (who can red flag / mod flag) > 10k users (who can vote on flags) > room owners (who can kick, timeout, move messages, etc. in their room) > diamond moderators (who can suspend, delete messages, freeze / delete rooms etc across the server) > chat moderators (who have all of the capabilities of moderators and room owners across all three chat servers) > staff.