Most of this is irrational fear of the "what if" which is causing you to make flawed decisions. Most of these reasons listed are non-issues for the most part:
1) deal with your car not starting - 99% of the time it is due to a bad battery. Batteries last 5-7 years and usually die in the winter. If you want to avoid this issue change your battery after 5 years or wait until it dies, get a boost, drive to the mechanic and have the battery changed.
2) your brakes not working properly - Brakes are a wear and tear item. If they are worn and need to be replaced then you replace them. Worn brakes still work properly. Excessively work brakes make noise and cause extra wear on other brake components but still work properly.
3) car overheating on the highway - Cars have improved since the 70's. When was the last time you saw an overheating car on the side of the road? If a car overheats it is usually do to improper maintenance.
This is a nice feel-good response for people who have decided to buy used. Definitely "words of encouragement".
However, my statements were not based on "irrational fears", but rather real-world experience from my own used cars as well as used cars purchased by family and friends.
1. The battery may be the cause of cars not starting, but it isn't always the culprit. For example, a bad alternator can cause the battery to drain out. Bad connection cables can cause it to disconnect. So no, changing the battery is not always the solution.
2. Bad brakes aren't always a danger, but they are certainly a nuisance and can make driving miserable. I had TWO used cars that would stutter when braking at anything over 30mph, and it was miserable. Changing the brake pads didn't do squat.
3. Cars don't overheat the way they used to from "overuse", but they can overheat from other problems, such as the fan braking down or shorting.
Either way, these are just a drop in the ocean in comparison to the hundreds of possible problems with a used vehicle.
Back to my point, if you can handle a problem with your car every now and then, you will save a lot of money buying used. If you can't or simply don't want to deal with problems, leasing or buying new is the way to go.
As stated previously, you could get lucky and have a used car that gives you no problems for years, but don't count on it.