A common thread weaves through many of Donald Trump’s picks for his incoming administration, a quality the president-elect values as highly as loyalty and perhaps even more than conventional qualifications: a flair for television.

President-elect Donald J. Trump, whose rise was fueled by reality TV stardom, is once again turning to television to recruit the key cast members of his new administration. The latest was Dr. Mehmet Oz, the former syndicated TV host, who was picked by Mr. Trump on Tuesday to oversee Medicare and Medicaid.