There is an old adage, that says they won’t offer an extended warranty on products likely to need it and there is truth to that. As others have said, it really depends on your personal situation and how much does the warranty cost and what would it cost to repair. If you are generally cash poor, where you couldn’t lay out 2 grand to fix your hvac unit, then an extended warranty would be warranted for example. It also depends on how much you value piece of mind, too.
I will tell you on cars that these days if you maintain a car and don’t overtax them, major things like the engine and transmission don’t fail that often, when I have seen them fail it is because the people quite honestly were often responsible, like towing something at the weight boundary of the vehicle’s engine and drive train.
I also would be careful about who is offering it, those places that advertise on tv don’t have a good reputation, and there is fine print that needs to be looked at, like "proper care’ (ever lease a car, turn it in, and you find all these things that aren’t normal wear and tear…same idea).
With things like my heating system, ac (when I had central AC) and the like I have a service contract through my utility. Unless it is a major issue like the heat exchanger cracked, they do the work either for free or for a nominal charge. It isn’t much, and generally they have been pretty good.
If a unit has a reputation for being crappy likely they won’t insure it, or if they do it will be a lot. Like any kind of insurance, they use actuarial tables to figure out expected losses , and that is how they make their money.
I think if the plan is good (which is the clinker, if it is good), for piece of mind it is worth it. We have hard water, and with our Bosch dishwasher I got the extended warranty (I have found that if the unit is going to break, it is likely just outside the main warranty, and if it then keeps working, if it is likely to break, it will be before the extended warranty ends).
I don’t think it is necessarily stupid to have insurance, we do on our cell phones and I get apple care on computers. The display on Mac computers have a tendency to have problems with burned in images and other issues because of the Iris display.
We do try and have reserves to pay for repairs, and for us in most cases it has worked out well. I also try and do research on products in terms of quality, like the heating units we have, and about repairability. For example, I would never recommend a Lennox unit , not because they aren’t good, but because they use proprietary parts that are hard to get and expensive to replace. I am careful with things like cars to get them regularly maintained, even if people think a lot of it is a waste, and so far it has paid off for us, the only major incident I had with a car was when my garage got invaded by rodents and they did unspeakable things under the hood (to the tune of 15k), but because it was a new car with full coverage, insurance paid for it.