I don't think they can simply deny OJI/disability for mundane tasks around the workplace even if it's not part of your job. If you were instructed/trained to avoid something inherently dangerous then they might have a leg to stand-on. I might be more concerned about accidentally dumping the wheelchair pax on the floor and getting in trouble if the pax has a liability claim.

Non-mundane tasks like fueling an airplane or operating a belt-loader might be different since you clearly need training for that, and you have no authorization to do it.

Of course if the company creates a climate where you're expected/encouraged to do tasks you're not trained for then they could be a world of liability hurt for NOT training you... that could apply if it's common for pilots to be helping out on the ramp. They have a duty to either train you or ensure that you don't do dangerous work.

I can assure you that Uber provides liability insurance for pax for all of their services, they could not possibly be so stupid as to rely on their pothead drivers for that. They may not insure the drivers for comprehensive, etc. So the driver might be screwed if he's doing uber on his regular private citizen insurance, but the pax will be fine.

All that said, if you hurt your back pushing a wheel chair or handling baggage, I'd probably just say it happened climbing out of your seat in the cockpit... nobody would argue that i your flying a 737

Not "no insurance", I doubt any state allows that. But many have comparative negligence, where the "victim" is responsible for some of the damages if his own actions aggravated the situation... such as not wearing seatbelts. I wear them always.