Assuming they could even amass great wealth (this is much more dubitable in market anarchism, where there is no state to aid the rich become as wealthy as they do), any rule they tried to impose would be seen as illegitimate. You've already raised a good counter-argument with private courts.
A capitalist society is the only stable society as the wealthy and powerful are completely at the mercy of consumers along with everyone else. In our mixed society the wealthy and powerful have an ally with a near monopoly on voilence, the state. The wealthy and powerful, and the state require eachother to exist. The state creates monopoly and protects the wealthy and powerful from competition through the use of regulations. The wealthy and powerful use their influence to promote the state over civil society.
Without a state owning a monopoly on violence the wealthy and powerful find themselves at the mercy of fickle consumers whose preferences for goods are constantly changing. The wealthy would face competition from entrepeneurs trying to get consumers to buy their products at the expense of those of the wealthy.
In a world where security is provided privately, it is least expensive and most profitable to keep one's protected well armed, which implies that they can easily fight you if you decide to become a power over them.
The fallacies of intellectual communism, a compilation - On the nature of power
In my view of history, the opposite is true. That is the wealthy have used their influence over the state security appratus to stop those seeking power over them. The best examples of this are the Communist Revolution in Russia and the Facist Revolutions in Germany and Italy. In these cases small groups seized the power of the state and use it brutally over their competitors.
In the mixed economy of the US in the early 1900s, much freer than it is now, the Standard Oil Company seemd to have all the force, money and control in Washington. But when the Washington decided to rebel and break up Standard Oil the company had slightly less than a 60% market share.
Point to ponder: Who says you gotta admire the wealthy?
(In an anarcho-capitalist society, influence cannot come without an antecedent admiration.)