You answered your own question. Oswald was a nut. Nuts do crazy things. Only Oswald knows his exact subjective motivation. Whatever it was, it had nothing to do with the fact that some republicans in Dallas didn't like JFK.
It's ironic that the Birchers saw the hand of Communists behind everything and the JFK conspiracy believers see the hand of "the CIA" or the "deep state" behind the entire assassination. In this case, Witt was part of this conspiracy. One side saw JFK as a traitor; the other sees everyone else including Ruth Paine as traitors. "Everyone else" being a figure of speech.
Oswald meanwhile was just some poor guy used in the plot.
As a side note: Bugliosi states (in "RH") that John Welch, the founder and president of the John Birch Society, denounced the assassination of JFK and expressed grief over the murder.