That is, when is chat-initiated typically a better fit than something initiated by a check-in, build, incident detection, and so on?
(Not knocking it, just curious)
Chat is also a great environment for getting notifications about extraordinary events. We get notifications in chat if a link on our homepage is 404ing, whenever deploys happen, or as scheduled updates for things we want to pay attention to but don't want to manage a separate website to handle them.
We usually start with small commands and then grow them to automate more of the "well trodden path" as we understand what we want to automate. Because every command (we call them skills) can expose an API, it's possible for an Abbot skill to respond without any person being involved.
A big benefit is that you don't have to share credentials from the Twitter account, but still make it available to people in your company.