Unscientific observation but most of the first generation JFK skeptics were Liberals. Since the 1990s, the Rightwing seems to have more Skeptics and Conspiracy Theorists. The biggest rightwing CT?er is now Donald Trump.
Mainstream/Establishment Liberals who work for CBS, the NY Times and other traditional Media have always supported and defended the Lone Assassin narrative.
I said left not liberal. And also the most noted figures not ordinary people.
The main figures - the most vocal or noted people - have been people on the left: Lane, Oliver Stone, David Lifton. Garrison is the odd one here since he called himself a conservative/libertarian. Lane and Stone are leftwingers not liberals.
What conservative/right JFK conspiracy is there today? Ventura? I'm not sure you can place him on the political spectrum. He's neither right or left.
Your point about the news media is spot on; I think the criticism from the CTer side that they were too believing of the WC report is pretty accurate. The 1960s coverage by the media was too pro-government; Vietnam and Watergate changed that.
With the risk of sounding like I'm defending the man, Donald Trump has stated that he believes Oswald alone killed JFK. Sure, he smeared Cruz's father but that's how he operates. He'll say anything to attack his opponents; most of which I don't think he believes. The man has no ethical standards whatsoever.
As to conspiracy belief today: the rise of conspiracy belief, it seems to me, comes out of the extreme partisanship of the time. Bot the left and right embrace all kinds of conspiracy thinking if they can use it to attack the other side. It's all mud slinging now; no sense of decency at all.
The best example of this is the collusion theory: the left believes Trump conspired to Putin to affect the election; the right believes Trump is a victim of a "deep state" conspiracy that is out to destroy him. Both side don't care at all about promoting the theories without much evidence.