A human chief of staff makes sure the owner's decisions turn into finished work: they take a request, work out who and what it involves, follow it through, and come back when it's done or needs a decision. An AI chief of staff is useful to the extent that it can do the same across the systems the business runs on.
Most of what gets written under the name describes something narrower: an assistant connected to email and calendar that summarizes the inbox, prepares a morning briefing and drafts replies. That helps an owner whose day runs through the inbox, but it stops at telling the owner about the work. It can report that a proposal is overdue. It can't open the CRM notes, write the proposal and put it in front of the owner to approve.
The chief of staff I install is one agent connected to the website, ad and social accounts, analytics, search data, CRM, ERP and accounting, and email and calendar, with a browser for anything without a connection. It does what an AI executive assistant does in email and calendar, then carries on into the CRM, the ads, the website and the books. It runs on the company's own accounts and keeps standing instructions about how the business works: who handles what, which pricing rules apply and what needs approval. Given a request, it does the work in whichever systems are involved and reports back with the result and anything it couldn't settle on its own.
It doesn't decide what the business should do, build relationships with clients or staff, or make the judgment calls an owner would hire a senior person to make. Its job is to carry out decisions and to put the facts for the next one in front of the owner.