At that time, Dallas was a very conservative big city believe it or not. True to their genes, the media and democrat politicians immediately blamed a “right wing” mindset for the assassination.
If you watch the media coverage the day of the assassination, they did several “man in the street” inteviews to get peoples’ reaction to the assassination. They almost all said they thought Right-Wingers did it, and the interviewers were basically feeding them the lines. So the media was definitely in on “The Right-Wing did it!” meme.