Some channels have dozens of moderators, especially larger channels with more organised moderation policies/teams (although these channels are likely to already have dedicated moderator tools/dashboards like those which Barry has described). It would be bad security practice, and potentially a bad user experience for the viewers of the channel, if a mod could give any app permission to use these API endpoints.
With the current way the API is designed the broadcaster can grant the apps that they specifically want to have access, and then those apps may or may not decide which other users have access. Some broadcasters may want all mods to have access, or no access, or just specific users (regardless of if they are a mod or not), that’s all possible right now, and that granularity of access would be lost if any mod could grant access to any app, unbeknownst to the broadcaster.
If you’re insistent on using Chatty and refusing to use any other service then you’re obviously going to be limited by what Chatty offer. Chatty is primarily a chat client, and is not designed to be a feature rich moderation tool.
If you are willing use other 3rd party tools but still wish your mods to interact through Chatty itself then you could create a bot that the broadcaster grants permission to, and then the mods interact with that bot through Chatty which will perform the API requests. Or simply use a dashboard style approach that many channels/apps already do, like Barry has suggested.