Vincent Salandria remarked that when he gave a talk on the assassination that he would often get a more receptive response from a conservative audience than a liberal one. This was roughly the 1970s era. That's pretty anecdotal but I do think it reflects the idea that there's always been a populist Right that was more open to believing in a conspiracy than the Establishment Right. It's not new. But that's true across the board. It's more of an Establishment vs. Outside split than a Right vs. Left. Historically, it's been Left conspiracists who grabbed the headlines - Mark Lane et al. - but now we see the populist Right figures like Carlson (he is literally nuts, right? what's the explanation?) promoting it.
Still I think this is a more recent change on the Right.

It's all about Trump.
Pre-Trump: The Boomer/1960s New Left generation tended to be more conspiratorial and skeptical of institutions like the FBI and the military. People like Oliver Stone and Michael Moore were highly regarded in Liberal pop culture.
Trump's arrival into national politics has scrambled things a bit. He has made it acceptable for Conservatives to question US foreign policies, institutions like the FBI and CIA, and the military industrial complex.
And due to political partisanship, many Boomer, MSNBC watching Dems (even those who were part of the New Left in the 60s and 70s), have grown more fond of the national security institutions and foreign policies that Trump is perceived as a threat to. Conspiratorial views have also been shunned more in Democratic circles under Trump. There's not as much space on the Left today for anti-Establishment politics (but that may be changing soon depending on what happens with the NYC mayor's race).
How this ties into the Kennedy assassination is, regardless of who Americans hold responsible for Kennedy's assassination the universal view among those who believe there was a conspiracy is that our government lied to us about the facts of Kennedy's murder. To accept that our government would lie about the murder of a US President requires some level of anti-Establishment political beliefs. And as I said earlier, at least in the Trump era, the anti-Establishment politics seems to be more prominent on the Right today.
During the recent JFK assassination hearings in the House, as a Progressive, I was saddened to see Congressional Dems express indifference towards the new JFK assassination files while most Republicans at the hearings expressed genuine interest in the new information. Still, I don't think members of Congress accurately represent the views of their constituents. There are still many Democratic voters who believe there was a conspiracy in JFK's assassination. But among the Democratic and Liberal establishment, anti-establishment views and conspiratorial stuff are mostly shunned.