How do we hand off a customer without making them repeat themselves?

You have learned to escalate at the right moment. That is not nothing — most people either hold on too long or bail too early. But the customer does not experience your judgment. They experience whether they have to explain themselves twice. Be honest about why the handoff breaks down. It is not because you lack a template. It is because writing the summary takes effort right when you are relieved to be rid of the problem. You tell yourself the next person will ask if they need more. They will not ask. They will ask the customer instead. This is a team failure, not a private one. One person writing careful handoffs while the rest do not still leaves the customer repeating themselves half the time. Fix this together or it stays broken.