If every reply needs a person, that person has to be reachable. A native Mac app for the people who live in the inbox, and an iPhone app so a customer waiting on a human is something you know about in seconds rather than find tomorrow.
The full inbox in a real window — not a browser tab pretending. A count in the menu bar, a notification when a draft needs approving, and the whole thing keeps working while you are between wifi and a hotspot.
The count you actually need is how many things are waiting on you. It sits in the menu bar, and the list is one click away.
Push when something needs a person: an approval waiting, a customer asking for a human, a freight quote to price. Read the thread, see what the assistant proposed and what it cited, and approve or reject without getting to a desk.
One email. Not a newsletter.