Why are our bodies not evolved to store only limited amounts of fat, which won't kill us with all diseases associated with obesity? Wouldn't obese people die faster and clean the gene pool for people who are slim?
"Wouldn't obese people die faster and clean the gene pool for people who are slim?"
If your hypothesis is right, the selection for such traits would only have begun in the last 20-30 years ("Worldwide obesity has more than doubled since 1980" http://www.who.int/mediacentre/factsheets/fs311/en/)
Natural selection wouldn't have had time to select against people showing tendency for obesity.
Also, obesity is a lifestyle choice, and not merely predictable by genetic disposition. I could be a "fit" person, but genetically predisposed to obesity.
Look up also genotype vs phenotype for further reading (http://evolution.berkeley.edu/evolibrary/article/genovspheno_01)
For most of human history the excess of food has not been a problem, rather the lack of food, and subsequent starvation has been a problem. So genes that promote energy efficiency and fat storage have been very much selected in the human population. So much so, that humans are rather fat for apes.
As for the problem of obesity, it has been a rather recent problem, no more than one or two generation. Evolution works in generation time. So there is not enough time to weed out genes that promote obesity which would be pretty common in human gene pool.
It's very complicated to know how a trait would have been selected for or against in the ancestral environment. Tendency to gain weight does seem to have been selected for in some populations - e.g. pacific islanders. Presumably the decreased chance of dying from starvation was enough to compensate for any increased risk of heart disease, if in fact health risks did exist. Morbid obesity as we know it today may simply not have been possible without the modern diet, and modern society/technology to provide an endless supply of easy to reach, unhealthy, addictive food.
This is an interesting question, and shows good insights in trying to glean cause and effect of large social phenomena. However, the natural selection process as presented in your question has a couple issues.
Primarily, natural selection is a passive process. The concept that we developed those traits we "need" is a misrepresentation of the process. It is more accurate to say we adapt to take advantage of what we have developed.
I.e. Fish don't grow lungs because they want to go on land. Transitional fish go on land only if they happen to have lungs. If it gives them a fortunate advantage, then reproduction may expand the population of that attribute.
There is also the issue that obesity does not cull the population before reproduction in most cases. The effect you're expecting is less impactful then you assume in the OP.