There are other professional basketball leagues that don't have salary caps, and LeBron doesn't play for them. The NBA salary cap is close to or exceeds what he could make in other leagues. Additionally, playing for the NBA gives him greater exposure to the American market.
Do you have any countries in mind that are going to pay executives more per year than the US, even with a salary cap? My understanding is that the US is already so far above the rest of the world that it would have to come way down to lose out on a raw compensation basis.
IMO, a cap would mostly serve to rein in the well-networked, measurable-performance-leveraging aspects of CEO compensation. Compared to every other employee, CEOs have a far greater ability to extract money from a company, mostly independent of their contributions to that company. Is it any surprise, then, that they leverage that ability to earn far more than their coworkers?