The Tesla Inc. board convened Thursday at the company’s factory in Fremont, Calif., in a conference room where Elon Musk often spent the night. His sleeping bag was still on the floor.

The agenda was Mr. Musk’s audacious plan to take Tesla private, which he had sprung on the investment world 16 days earlier via tweets. He was joined in the room by board members, lawyers, bankers and advisers, members of a Wall Street transaction machine put into motion by the entrepreneur’s idea.

Mr....