That is not enforced by capitalism, though. Capitalism isn't a single incarnation of an economic system, it's an entire landscape of systems. Crony capitalism isn't the be all and end all - just look at the Gini-coefficient worldwide (https://en.wikipedia.org/wiki/Gini_coefficient); there are plenty of capitalist countries that do quite well.
> When I say "structurally enforces", I mean that it amplifies the natural urge to have more, or do better, than someone else.
And you base that assessment on what exactly? I happened to have experienced two worlds (communism and capitalism) first hand and I can tell you for a fact that no "amplification of urges" was necessary to have people striving to have more than others.
The main difference is that in communist countries common folk had no way of (legally) getting access to most consumer goods (apart from food and basic supplies).
Corruption, stealing (mostly public property, e.g. from state-owned factories or farms), moonlighting, and under-the-counter sales were the norm and overall everybody suffered the consequences of an economy of scarcity.
This however, didn't stop party officials, former squires, factory directors or high ranking government officials from owning big houses and villas, driving western cars, and having access to luxury goods that were unavailable to the general public (both domestic and imported from the West).
The main difference between capitalism and communism that I experienced are a higher standard of living (in capitalism), less corruption, actual opportunities for those who seek them and much less state sanctioned despotism.
There's downsides, too of course, but I live in a country where it's incredibly difficult (and somewhat frowned upon) to get obscenely rich and many of those that are inherited their wealth (something that could be easily fixed by 100% inheritance tax for sums exceeding some still generous threshold).
That's why I don't see mega-rich people as general problem with capitalism (it's more of a result of a global economy and financial markets anyway - the bulk of most billionaires' wealth is in stocks and options, not actual cash or physical possessions).
Just to give you some perspective on how "shit might go down": there was this guy, Markus Persson, who programmed a computer game that became the best selling computer game to date (200 million copies). He became a billionaire when he sold his company and the rights to the game in 2014.
Then there was this other bloke, John Carmack, who worked on 42 games and created genre-defining masterpieces like Doom. His games sold well, too, but he was 30 years too early and thus "only" became a millionaire (~$60 million).
The difference? Opportunity! Back in the early 1990s there was no internet and (PC) gaming was a niche.
I guess the lesson here is that the size of the market has changed and getting obscenely rich relatively quickly is easier than ever before. But that's just a direct consequence of the average person being wealthier than ever before and thre being more capital around.