Social Security has taken center stage after President Donald J. Trump told the nation it’s a system that has “shocking levels of incompetence and probable fraud.”

Although federal officials have long recognized improper spending as a problem, it is not the main reason for the program’s dire financial outlook

The involvement of Antonio Gracias, an investor who has known Elon Musk for two decades, suggests that Mr. Musk has made overhauling the Social Security Administration a priority.