WASHINGTON—Just days after Kristi Noem took office as President Trump’s head of Homeland Security, she accompanied U.S. Immigration and Customs Enforcement officials on a predawn raid of several neighborhoods in New York City. “Live this AM from NYC. I’m on it,” she posted on Jan. 28 at 4:43 a.m. on X, with a photo of herself sporting an ICE baseball cap and getting into a car. The problem: The raid was still ongoing when Noem posted about it, undercutting the element of surprise, according to people familiar with the operation. Noem’s handling of that early raid was emblematic of the...