Competence is the ability to do something (a goal) successfully or efficiently.
IQ measures the ability to perform abstract reasoning tasks successfully. Conscientiousness measures the willingness and desire to perform one's duties well. Additionally, among people that earned their wealth, the fact that they are wealthy demonstrates that they achieved their goal of becoming wealthy. These all indicate competence to me. Wealth, in general, can also be imagined as a state where one's needs are met, which is (for people that earned their wealth) almost always closely related to having achieved certain goals.
I would have a hard time imagining an argument that rejects these claims and also demonstrates that wealth and competence do not correlate positively.
Bear in mind that this is talking about populations as a whole- correlations are something we can tease out via sociological studies, and experiments are few and far between. There are facts about human nature that we will never prove conclusively, but to be an effective agent in the world, you should try to learn things about people that cannot be proven. Being too afraid to generalize the wealthy as competent causes yourself to miss an personal actionable moral lesson about goal-oriented behavior. It is entirely possible to make this judgement, and learn from it, without demonizing the poor as incometent or subhuman.