No one doubts that those were JFK’s words, any more than they doubt that Abraham Lincoln’s words or Franklin Roosevelt’s were his own. Why would anyone even bring up plagiarism? It isn’t an issue.
As for Fox News, it’s a recent phenomenon. The channel was launched in 1996. There was nothing like it in 1960. In those days of network broadcasting, all three commercial television networks (NBC, CBS, and ABC) had real news coverage with respected anchors and journalists. “Educational television,” the forerunner of PBS, was just barely getting started then and wasn’t doing regular newscasts. None of them was doing anything remotely resembling one-sided political propaganda or attack-as-entertainment.