I think the easiest way for someone to believe this is to give you a few situations that I have personally seen happen. (We run a bed and breakfast and see a large amount of travellers per year in a very touristy area of the world, Cusco, Peru.)
Situation number one I call no fun. Two really cool gals came to Peru on a pretty long trip. The basis of their trip was not Machu Picchu like most people, but the jungle, the Amazon Rainforest. The day before the trip to the Amazon the doctor came to the house and gave the bad news that the girl could not travel to the jungle where the medical facilities were nil to none, (she had picked up a bad case of Salmonella), and where she would suffer even more form the heat and dehydration. They did not have any kind of travel insurance and lost the entire trip. On a lucky note their company allowed them to use the trip at a later date, which they did.
Situation number two which was not pretty. Bucket list killer, Machu Picchu, the main reason anyone comes to Cusco. Every once in a while we get people here for a seriously short visit… arrive, see Machu Picchu, leave. Cusco is well known for their Strikes. “Viva el PARO!” These strikes by the people shut down everything. Everything..including the airport and all transport. Good luck getting to Machu Picchu on a strike day. Your Machu Picchu trip with nothing included except Machu Picchu entrance and the train will run you about 200-300 USD. (Not to mention flights etc.) She missed her trip to MP and all the money spent on it as well.
Situation number three, what a breeze. Just recently a plane had a small boo boo on the runway. Flat tire, hit a bird, who knows. It went off the runway, fell over and created a big stir in Cusco and shut down ALL flights in Peru for one day and all flights to Cusco for three days. A nice couple from Australia that bought a huge package from us for 10 days missed the first 4 days of their package. Adventure tours and part of their Machu Picchu trip were in those first few days. They had bought a travel insurance plan for somewhere around 100 USD pp for a month of traveling around South America. We went though their package and I wrote them receipts for their trip along with a letter stating everything that was missed, all the extra things needed like NEW train tickets and extra transports, and sent a letter to their company . They got back all their money.
Oh yeah and don't forget strikes by the people. When the people of Peru decide to strike it can last days on end and ALL TRANSPORT gets shut down. If you have a flight you will need to walk there. If you have a train to Machu PIcchu, sorry, but you will miss it. One year we had about 4 strikes a month. This year, 2017 we had months long of off and on strikes. Imagine the headache!
It’s a no brainer. Get the insurance.