Yep - did something to my knee in Majorca a few years back - paid for by the insurer, a company called Kingfisher who were just the one that the package holiday co (and I can't recall who that was, may have been Cosmos) used.
Since then when we went to Oz I claimed on the travel cover we got nominally free with our Lloyd's Bank Platinum account - but the actual co was AXA Insurance and they were fine too.
Both of those, we did have to do a lot of it ourselves, but to be honest no different to what we'd have done here. Did my knee, saw the GP in resort - he said I'd have to go to the clinic in Palma, fortunately we had a hire car but we could have taken a taxi, got a receipt and claimed the cost back if we hadn't had the car. He looked up the list of insurers he had in his drawer and told me which one they used so we just got the hotel to locate it on the map of Palma for us and off we drove. Handed over the policy when we arrived, they took all the details before passing me through to see a doc and it was I presume settled direct with them. I mean here - my husband or someone would have had to drive me to A&E so what's the difference? Difference is if I'd needed a taxi here - I'd have had to pay for that myself! LOL