This is way different than starting out with nothing like the US did and earning capital through hard work, before oil was even discovered. The US did not have the advantage of a huge supply of unearned capital to work with.
If you say the UAE has money because of capitalism, you would also have to say Venezuela has money because of socialism, when neither is the case, the reality is they won the oil money lottery. The poor in Venezuela have more now than they did before Chavez, would you say that is because of socialism? No, it is because oil money is greasing their economy to the point the poor have more, their standard of living has went up, which is why Chavez is so popular amoung the lower class there.
The US was capitalistic from the begining and unlike the nomadic ME countries was wealthy before oil.
If oil had not been discovered in the ME would the UAE have become a "capitalistic" country?? I highly doubt it as they had not embraced capitalism up to that point.