I think NAF1138 is right, that this is a case of a somewhat “inside baseball” term becoming well-known to people outside the industry, which came about as a result of the people holding that position becoming more well-known.
“Show runner” is not, and never has been, a term that you will find listed in the credits. People in that position will generally be credited as a “producer,” or some variation thereof. But there are a plethora of producers, executive producers, supervising producers, and so forth in the credits of any given TV show. It’s likely that the term “show runner” developed to identify the one producer who is really in charge of everything.
When I was young, it seemed to be fairly rare for anyone outside of the television industry to pay much attention to who was producing a particular show. Maybe part of it was my own lack of curiosity about such things, being a kid and all, but I certainly don’t recall my parents or any other adults having intense conversations about it when MASH* or Kojak got a new producer. It just wasn’t something that the average viewer took much interest in.
These days, we all seem to be a bit more aware of the behind-the-scenes aspects of our favorite shows. At the same time, long form arc-related storylines have become more the norm (even in sitcoms), so a change in producer is perceived to have much greater potential impact on the long-term progression of the show. I think that those factors have resulted in the insider term “show runner” becoming more familiar to general viewers.