Tyler Hassen’s initiation into Elon Musk’s Department of Government Efficiency, or DOGE, started as a series of tests.

The Houston oil executive says he was given vague instructions to show up at an unfamiliar D.C. address. He hastily packed three suitcases and flew east, taking an Uber to a building that he soon learned belonged to Elon Musk’s SpaceX. He was whisked to a private office, acutely aware that his tailored suit stood out among the hoodie-clad engineers “too locked in” to their laptops to look up. He was ordered to go through the website of the Department of Commerce and, within days, find two to five ideas to present to Musk.

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He checked into a nearby J.W. Marriott and told his wife he didn’t...