Now instead of "league" and "LeBron," substitute in "country" and "the world's best corporate executives" and you'll see why it's a bad idea. We want the world's best here.
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?
Superstars have a far bigger impact on their team's performance than an average player in the league. Many commenters have argued that good CEOs have large impacts on company performance.
The average NFL career is 3.5 years; the average NBA career is 4.8 years. Several commenters have pointed out that CEOs of failing companies risk this being their last CEO job.
Athletes are derided if they cannot save enough to ensure their retirement when they are making millions of dollars a year. Why can't we expect CEOs to manage their own finances responsibly and plan for a future where they aren't earning millions?
After their playing careers, many athletes choose to work lower paying jobs as assistants, coaches, commentators, consultants, etc. Why can't a CEO take a step back after a failure and be a COO, CMO, VP of Sales, etc.?
During their careers, NFL and NBA players' earnings are limited by a salary cap. LeBron James can't earn more than ~$30MM from his playing contract, even though he generates far more revenue in TV contracts, jersey sales, gate receipts, etc. Why can't we impose salary caps on CEOs?