Wealth inequality is a consequence, not a goal. No political system in the world explicitly takes action to increase wealth inequality, their stated goals are (depending on ideology) to either decrease or ignore it.
Since the conspicuous examples of wealth inequality in functioning states usually involve the latter, we can consider: Why wouldn't a government reduce inequality? Or alternatively, why would they tolerate it?
The first benefit is stimulating innovation. The people at the top have it really good. A lot of other people want to be like them really bad. They will try any perceived avenue of getting there with much enthusiasm. If the market is functioning reasonably well, the chief ways of getting very rich would involve somehow improving the economy as a whole: Whether by making the whole system or part of it more efficient (thus collecting profit from arbitrage), or by inventing some new service or product that answers a previously unsatisfied demand. Thus everyone benefits from the market being more efficient, by being able to buy nicer things for less. Of course, this breaks down if there are perverse incentives. For instance if corruption is a viable and profitable activity, many will try to get rich by this parasitic activity rather than something that benefits the economy. Or if rule of law is poor, crime lords will proliferate and prey on citizens. Moreover, it must actually be possible to innovate and compete in the market, the government for instance must not bog down new companies with red tape.
Without inequality, there are still other motivations to innovate, like personal satisfaction, desire for fame or legacy, and a sense of civic duty. But the motivation of wanting to be very rich is gone. Thus it is not accurate to say that no innovation would happen, but we could expect that less of it would happen in proportion to the lack of inequality. In practice, this has occurred to some extent. For instance the selection of products available to consumers in the USSR was notoriously small (disregarding their availability, which was also poor) compared to the West where often countless different competing products or services are available, and it was difficult to find high quality products (while in the West you have the option to pay a little for poor quality, or pay a lot for exceptional quality, and everything in between).
The second benefit is that everybody tries to do their best. Obviously few want to end up at the bottom, so no matter what job you do, you will try to work really hard and do your best, hoping to get a promotion, a raise, a bonus, or in the least to not get fired for poor performance. Again, this requires that good work actually be rewarded to some extent. If there is a widespread culture of nepotism in every major industry and hard work is never rewarded, then this point does not apply.
In more egalitarian systems, such as the USSR, there is a problem of getting people to work. Again, some people will work for non-economic reasons, but there are others who won't bother unless they have something to gain (or lose). Thus egalitarian systems often punish people who don't work hard enough. You could say that under inequality, productivity is maintained with the carrot, and under equality it is maintained with the stick.
Lastly, and probably most controversially, it allows society to remove or minimize the least productive elements of society. For obvious reasons, proponents of less-egalitarian systems will downplay this, but it is inevitably an important economic consideration. In any egalitarian system there will always be people not pulling their weight (due to inability or unwillingness), and each such person is essentially a drain on the collective resources of society.
In an egalitarian system, such people would be entitled to some decent standard of living, so it might be necessary to ensure that they have nice houses, eat nice food, send their kids to good schools, can get quality medical treatment, etc. Of course someone has to pay for all this. Arguably many people, when provided with a decent life, will become more productive - but not all. If the system is less egalitarian, the least productive people are not expected to receive these niceties, typically they will at most get a basic food allowance and a bed at a homeless shelter. This is naturally a lot cheaper, though arguably less ethical. In support of this, tax rates tend to be much higher in places with less inequality, while the standard of living for the poorest people is higher.
So it's some sort of "pyramid scheme", where everyone rises, but those beneath the top are being scammed.
In a pyramid scheme, most people involved get less out of it than they put in, while being promised otherwise, hence the scam. A few rise, everyone else falls. If everyone rises, how can it be a scam? Sounds more like a win-win to me.
One could argue that people are being scammed in the sense of not rising as much as they could have. But in many systems with inequality (assuming they function well in the sense of the three points above), you can obtain a pretty decent standard of living with a reasonable amount of work, so I wouldn't call that a scam. In the US, for instance, median household income is $60k - which while not extravagant, certainly would not leave you starving in the streets.
In egalitarian systems, the equality is often achieved by confiscating part or all of the income of people who produce the most value, and then distributing it to those who produce the least. These people get less out of the system than they put in. The very bottom stratum places a disproportionate strain on the welfare system, getting free things at the expense of everyone else. If anything, this seems like a better candidate for a pyramid scheme.