Live chat is not a desk
A shared inbox is the right tool when the customer is still in the widget. It is the wrong tool for a refund that will take two days, a bug that needs an engineer, or anything that should survive the visitor closing the tab.
That is the gap tickets fill. Handoff keeps a person in the thread. A ticket takes the work out of the thread and gives it an owner, a status, and a number the team can find on Monday. The two are siblings, not substitutes. See human handoff for the live path.
Use the inbox while the customer is waiting. Use a ticket when the work will outlast the chat.