WASHINGTON—Kellyanne Conway, one of President-elect Donald Trump’s top advisers, scolded the media and bid good riddance to the old Republican order at the WSJ CEO Council gathering on Monday night.

Mrs. Conway, one of Mr. Trump’s most effective messengers, did it all while smiling from her seat on stage—winking at the crowd—and saying the signals for his stunning victory were written on the wall for everyone to see.

“People...