I'm a former travel rep so could fill an entire thread with stories of travel customers having accidents and needing to claim on their insurance. Some examples:
The chap who broke his leg skiing in Canada - the rescue off the mountain, the 5 days in hospital, the ambulance transport (to the hospital and later to the airport) and the operation itself would have cost tens of thousands of pounds.
The man who broke his shoulder falling off a mountain bike in the Alps - the EHIC covered his medical bills, but his travel insurance covered the cost of flying the whole family home at the end of the holiday (his wife couldn't drive) and the transportation of his car back to the UK.
Another broken leg due to skiing one - a young kid broke his leg and one wrist in the fun park and couldn't fly home for 10 days. Insurance paid for him and his mother to stay in a hotel in the resort until he was fit to fly, then paid for their flights (the family had originally driven out).
Finally, the lady who having endured a 16 hour coach journey to the Alps slipped on the steps outside the hotel on arrival and broke her femur, requiring a couple of operations, several weeks in hospital and a repatriation flight.