The reason that few people can withstand unforeseen expenses is that they never plan for them. Insurance is now a necessity, and any enormous expenses that Americans are faced with is going to be covered by their insurance. If Americans weren't so insurance dependent, then their expenditures would drop in order for them to form savings in case of emergencies. The questions of how to deal with $1000 emergency room visit or a $500 car repair is simple; don't go to the emergency room, don't buy a car.
Some people will certainly be put in a situation where their life is on the line, and an emergency room is just what they need. The issue is that this is really a very rare occurrence, and many people go to the emergency room for non-life-threatening conditions (e.g. getting stabbed in the arm, getting concussed, even losing a tooth). For most people, an emergency room visit is not necessary. For most people, a car is not absolutely necessary. These expenses can be avoided by understanding that a car isn't worth the capital and maintenance required, and that by your own genetic resilience you are capable of surviving an awful lot.
Materialism is growing in the United States, though, which means that having a car is a necessity if you want to be top dog in your social circle, and recuperating on your own from stab wounds is for poor kids in the ghetto. Budgets do not account for trouble because it's not cool to need a budget and because insurance covers most of our unforeseen expenses.