Big companies already have elected boards.

Big boards of shareholders consisting of the CEOs of other companies. Its how you get industry cartelization.

Why do CEOs reciprocally sit on each other's boards?

The reciprocal interlocking of chief executive officers is a non-trivial phenomenon: among large companies in 1991, about one company in seven was in a relationship whereby the CEO of one company sat on a second company's board and the second company's CEO sat on the first company's board. We develop hypotheses to distinguish whether this practice furthers the interests of shareholders or the private interests of the CEOs. Using a sample of large companies, we employ a probit model to test these hypotheses. Our empirical findings are that these reciprocal CEO interlocks primarily benefit the CEOs rather than their shareholders.

Very often, a CEO will receive stock in-lieu of compensation. This makes them a major shareholder of their existing firm. Firms will also use stock in-lieu of payment when negotiating contracts between firms, particularly in M&A and other consolidation agreements.

Consequently, you'll have a guy like Michael Dell, whose primary wealth comes not from owning shares in Dell Computers but in Broadcom Semiconductor Company. This came about because he received 22M shares from Broadcom in exchange for his controlling interest in VMWare, a company he obtained by trading his Dell stock to the original owners.

He sits on Broadcom's board and the former CEO of VMWare sits on his board. When Broadcom skyrocketed in valuation (currently in the $1T range) during the Crypto/AI induced chip shortage, Dell's net worth skyrocketed with it.