Some comments on what people wrote. Note: to get the bottomline of what I'm saying, only read the texts in bold.
"I agree that capitalism is not "fair", but I don't know of any economic system which is "more fair" than capitalism. Do any exist? Have they been tried and proven?"
When you go to the bottom of it, the problem does not lie within the economic system or form of government; socialism and communism are absolutely great ideas: equal sharing of goods and commodities among everyone, it just doesn't work. Capitalism, in its written, ideal form, is also a fantastic idea: constant growth of economy, everybody is happy and do well, the sun shines, birds sing and children laugh. Then you wake up and go walk the dog. The keyword to understand why it just does not work is greed. I'm sure you get my point.
It really makes no difference how a government chooses to deal with wealth and the distribution of it. What does make a difference is how it chooses to deal with the people. Or, in other words: What makes all the difference is how people choose to deal with people.
"Singapore is a good example of a successful centrally planned economy."
While this may be true, Singapore has huge problems. The anti-drug policy of the country is among one of if not the harshest in the world:
(This is somewhat off-topic but, well reflects the way priorities are set in this "successful economy".)
"The death penalty is mandatory for anyone, over the age of 18, found in possession of more than 15 grams of heroin, 30 grams of morphine or cocaine, or 500 grams of cannabis."
That is, drug abusers, those in possession of drugs and minor traffickers are put to death while those behind the crime escape. Statistics indicate that out of 340 people executed between 1991 and 2000, 247 had been convicted of drug trafficking. The real problem here is that the situation only goes worse all the time. Also note that in Singapore it makes no difference as to why you were in possession of large amounts of drugs; if the abovementioned limits are passed, you will be executed even if the drugs were planted on you unless you specifically manage to prove yourself innocent. This is a perfect analogy to how people are sent to jail in the US. All the time the laws get stricter and harsher and the means used get more radical. I see this as a downward spiral - it will not work in the long run.
Even if this has nothing to do with capitalism per se, it does, in part, come to prove how concentration on economy can (and likely will) override or even exclude the caring for people. I'm sure everyone has, at some point, met a person with "so much money (s)he does not need to care about others, or how they feel". Blind, ignorant arrogance at its worst.
"Why is a rich person more worth than a poor person?"
The answer to this is very simple: the amount of money involved. Especially in a capitalistic system. There it is, even in the very name of the system: money. A rich person is exactly what you wrote: worth more than a poor person. This has nothing to do with humanity or fairness. It is no different from the fact that in capitalistic countries it's capita that is being protected, not the people. Money is the (only) real safety net in such systems. That's true hardcore capitalism all the way.
What makes me wonder is how anyone has the nerve to call the capitalistic systems sophisticated, or the people praising them as good intelligent or even human. A quote from Douglas Adams' "The Hitch Hiker's Guide to Galaxy":
"...Man had always assumed that he was more intelligent than dolphins because he had achieved so much... the wheel, New York, wars, and so on, whilst all the dolphins had ever done was muck about in the water having a good time. But conversely the dolphins believed themselves to be more intelligent than man for precisely the same reasons."
Amen to that.
A big thanks to geekaus for pointing out a typo in the wu that made nearly the whole writeup appear in bold.