A JFK conspiracy will always appeal most to both the far right and the far left. Because an extreme CT position will maintain that democracy was taken over in 1963 and we have since been controlled by a deep state, which uses the threat of assassination to maintain control of the government. If one can get people to believe this, their support for democracy will be greatly diminished. From there it is only a small step to say that no only does a deep state control us today but that a democracy will always be vulnerable to being taken over by such a conspiracy. If this is the case, one might as well accept an authoritarian government, which at least does not rule in secret. Both the far left and the far right would like his, provided the right people are left in charge.
It is always possible for a supporter of democracy to believe in a conspiracy. Particularly if they don’t hold an extreme view, that the conspirators took permanent control of the government in 1963 and have control to this very day. But the CT position will always be most appealing to those on the extreme left and the extreme right and who, deep down, don’t really believe in democracy.