The Sinclair Broadcast Group (Smith) and the Trump administration’s F.C.C. chief (Pai) see eye-to-eye on the need to unleash television. An examination of the F.C.C. records shows that the Smith-Pai alliance does not follow the familiar script of a lobbyist with deep pockets influencing policy. Instead, it is a case of a powerful regulator and an industry giant sharing a political ideology, and suddenly, with the election of Mr. Trump, having free rein to pursue it — with both Mr. Smith, 66, and Mr. Pai, 44, reaping rewards.