Elon Musk‘s stated plan to find someone to succeed him as Twitter Inc.’s chief executive officer has set the billionaire up with the thorny task of identifying a candidate for a job he has said isn’t desirable. Running Twitter would be a tall order under any circumstances. It was a consistently money-losing business before Mr. Musk acquired it in October, and is now in the middle of broad tumult in its industry. Meanwhile, many advertisers, Twitter’s main source of revenue, have stopped spending since Mr. Musk’s arrival, and he has made a rash of changes, some of which he reversed. Mr. Musk has a collection of Silicon Valley friends and allies who provided advice during the run-up to his acquisition or since he took over—though none have been publicly named as Twitter executives.

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