I have been proximal to the CEO search process at a few different large companies, and at a minimum have anecdotal insight into what drives selection.
CEOs are hired to fix perceived fundamental problems with the business, even at externally successful companies, and are evaluated for fit based on their history solving similar problems at other companies. Many times the business problem driving selection is invisible to casual observers, particularly those external to the company because it involves internal board/investor topics. The CEO's primary job when hired is fixing whatever it is the board deems it necessary to fix, not posting good numbers per se.
Fixing a company is exponentially more difficult the larger the company. The ability of most people to effectively reshape a company in some key way scales very poorly with company size. There is definitely a Peter Principle at work and most successful CEOs fail to be effective at some scale. To make matters worse, most boards have ambitions to grow the scale of the business, so they tend to want to hire CEOs with experience that exceeds the current scale of the business, which creates an intrinsic shortage of candidates with proven experience to reshape organizations at the desired scale. Supply and demand.
It is ironic that this site often has horror stories of working at startups -- tiny organizations -- with disastrous CEOs, but in the same breath people assert that anyone could be an effective CEO of a 10,000-person organization because the job entails no meaningful skill. Most people are demonstrably ineffective as CEOs in a 25-person organization, there is no reason to believe they would suddenly become effective if the organization was a hundred times as large.