WASHINGTON—During his first term, Donald Trump would sometimes interrupt meetings on military or veterans’ issues to call Pete Hegseth, a Fox News host who impressed the president with a blunt style that mirrored his own. Hegseth railed against Pentagon leaders on television and pushed pardons for soldiers convicted of war crimes, a contrast to the cautious career officials who frustrated Trump. For his second administration, the president-elect has nominated the telegenic 44-year-old Army veteran to be his defense secretary, despite—or because of—a résumé thin on government experience. “President-elect Trump might look at Pete Hegseth as someone who will carry out...