From recent experience last year of such an incident, I think it goes like this. The handler only has insurance up to a certain figure. The airlines accept this as otherwise the handling fees would be more. So if the costs come to more, it is uninsured. The handler is not liable, even if plainly it was negligence on the part of the staff. The airlines accepts in its contract with the handler that above the insured figure it carries the can itself. One of our planes was severely damaged in an appalling towing incident. The damage was extensive right back to the forward hold. It took 24 Boeing staff 10 days to repair in round the clock shifts. I believe the handler walked away from it.
The ground accident record at airports is stunning. That is where aeroplanes get hammered. Watching some of the groundstaff at work is excrutiating.