baconbaconpancakes
Tourist
Your answers are always spot-on and well thought out.There are many different ways to look at insurance companies being "worth it".
If you're looking purely financials, it wouldn't be much of a business model if they paid out more then what people pay in. They've got people crunching numbers to make sure the average customer doesn't get more then they pay. Some do, most don't, overall the company stays afloat. Because you don't know if your dog will ever require more costly care, it's hard to just look at the financial side. If he never gets sick, your 45$/month x 10 years is 5400$ you didn't spend on insurance that you could use for another dog. If he does get sick, you might spend more than that on care, and if it's early on in his life that he gets sick, you might not have enough funds saved up.
So, from a peace of mind perspective, it can take the issue of money off the table if something bad happens to your dog, even if that peace of mind might cost you overall throughout his life, that's something many people consider "worth it".
Also, depending on the coverage plan, it might grant you access to specialized care that wouldn't otherwise be realistic, like expensive chemotherapy, advanced diagnostic imaging, specialized surgery, dialysis or mechanical ventilation, which can be in the 10k$ and up range. Maybe those aren't common things, and they might not be for everybody, but if you have an emergency and they throw scary big numbers at you, it's one less thing to worry about.
Also, cost of vet care is growing fast, especially if your dog needs specialized or emergency treatments. When you consider the different coverage plans available, you should give yourself an idea how much care in your area costs. For example, if they cover 2000$ and emergency surgery is 4000-6000$, you may or may not be getting all the peace of mind you think you're paying for. You also want to make sure you understand if they're putting exclusions in your policy (like if the dog has pre-existing problems that they won't cover)
In this area, I'd ballpark hospital stay with surgery (like pyometra or intestinal obstruction) at a regular clinic is roughly 1500-2500$, and figure 4000-6000$ in big emergency center. That might vary from place to place, you can probably get some rough estimates for frequent emergencies by talking to your vet.
I do think it's really worth having insurance if you have a puppy, like first year or two. Overall cost for 1-2 years of insurance is nothing, and that's a time where you get to know if he has dangerous habits like eating socks, throwing himself down the stairs or chewing electrical wires, or if he gets development problems like bad joints that need surgery or stuff like that.
After that, it's "worth it" if something terrible happens, that part is really a personal decision.
The simple answer is that all dogs need insurance for expensive procedures. They will get old and need care. Either you are well-off enough to always have about 15k per dog available at all times and you are the insurance, or you are not and you should pay monthly for an insurance company.