I'd like to add a less cynical perspective: The average person doesn't "have the money to buy a house". Or even a car at that. They take out loans and plan on paying them off over several months or years. Using your logic, the average person should not buy anything unnecessary to survival seeing as they can't even afford their car or home. There is spend and live. And then there's spend, spend, spend. The idea is not to go overboard and do the latter.
You're suggesting that people prone to financial setbacks should not have any extras and just save their money for hard times. A lot of people save some of their money. But you never know what's going to happen. The car might break down. Someone in the family could get seriously ill. Think about even something as simple as the refrigerator being broken. All costly things. That's not even day-to-day situations. My brother accidentally dropped and broke a gallon of milk once. That's $2.50 down the drain. Oh well, should've bought milk insurance. You cannot possibly plan for the all of the worst scenarios in life. You just have to expect that there will be some and do the best you can.
Some people are not able to save very much money. But they do everything than can to make sure their pets' and children's needs are met. No one in the story expected the vet to cover the cost of the dog's surgery. The family knew they would no longer even have visitation rights to their dog. As for the for the people who have saved enough money to afford that kind of surgery, good for them. I'd consider them rich. The average person cannot afford to have $6,000 just sitting around in a bank account waiting for an emergency to happen. (By the way, it's not as if all the veterinarians out there have payment plans set up. The banks don't necessarily happily agree on a loan for a pet's surgery either. So no - a car versus a dog surgery is not comparably cheaper in the long run. Suddenly being in debt $6000 with interest? If it's not payed off right away, the bank may take away your car). Also, so what if someone has two emergencies at once? Let's say they had two dogs who had slipped discs. Should they have $12,000 saved?
I know what you're trying to say, but I don't think it's applicable in this situation. The couple in this video don't look like a dirt-broke couple. They're not wearing shoddy clothing. The woman even has sparkly earrings on. And if that's their home behind them, they have furniture. The average person can afford some moderately expensive things. I realize that's there's pet insurance. I've checked out a lot of them. For a nice penny every month, they'll cover pretty much what regular people insurance does. Which is not everything.