There is one grand fallacy in the model proposed here: it assumes too much. Cletus has done a good job of outlining most of these, but he missed one very important one:
We assume that everybody has the same utility function.
For this to be true, it would require that all people are able to perform every single known type of job, all at equal levels of ability. This is quite clearly false; different jobs require different levels and amounts of training, and in addition to this some people can simply perform a given job better than others can, for reasons that can't really be explained (but are often attributed to innate talent or simply blind luck). Perhaps the two most extreme examples of this are the fields of medicine and law. Both require years of training beyond even college, and there are very good reasons for this. I would not want a lawyer performing my heart surgery any more than I would want a doctor arguing the Microsoft case.
It should be pointed out here that if wealth were distributed equally to the current per capita income of the US (slightly over $30,000) no one would be able to afford medical school or law school, and thus there would be no doctors or lawyers. Since income is meant to be distributed equally, there would be no chance for loans under this system (as they would upsst the artificial "balance" of wealth). Whatever you may think of lawyers, clearly this scenario would be disastrous.
Another note: resources, even wealth, are finite. It is assumed that equal redistribution of wealth would raise the standards of living of the majority, but this is not true. The median income in the US (the point at which 50% of the people earn a greater amount and 50% earn less) is actually much higher than the mean, which is often called per capita income. So in the end, more people will suffer under this system than will gain from it. I doubt this is what anyone wants.
And one final thing: many here speak of "the gap between rich and poor" as if the current distribution of wealth were actually that clear. The fact is, it isn't. Our economy is not a pyramid with the rich on the top and the poor on the bottom; in the end it is more like a diamond. There are a very few extremely wealthy people at the top, and there are some poor at the bottom, but by far the most people are somewhere in the middle (and in fact, in the current economy the widest point of the diamond is somewhat above its vertical center). In other words, the distribution of wealth is not equal at this point, but it does follow normal distribution ("normal" being used in the statistical sense).
In the end, while this may not be perfect or ideal, it may well be the best we can hope for given the non-ideal world in which we live. It sounds callous, I know, but you cannot end the suffering of the few without causing the many to suffer. Given this, the best we can offer is equal opportunity to succeed, while acknowledging that this does not necessarily guarantee equal results.