Elon Musk aggressively elbowed his way into the space launch business over the past two decades, combining engineering genius and an entrepreneurial drive with a demand that the U.S. government stop favoring the big, slow-moving contractors that had long dominated the industry.

Today, it is Mr. Musk who is dominant. His company, SpaceX, is the primary provider of launch services to NASA and to the Pentagon. His rockets carry far more commercial satellites into orbit than anyone else’s, including those for his own Starlink communications network. He has set new standards for reaching space cheaply and reliably.

But in one striking way, the former outsider has come to resemble the entrenched contractors he once fought to topple: He is…