Jon Stewart spent most of his entire career railing against the Fox News, as he called it, "bullshit mountain", and there is very much evidence that its founder came from Russia with the express intention of making it that way. Or if not, then like Tucker Carlson, at least did not turn away free money when it was offered to him to act in a certain particular manner.
Though what I will point you to instead, b/c you seem like you will REALLY enjoy it, is the video series from Innuendo Studios called "The Alt-Right Playbook", which contains essentially the material from a college-level course in this exact subject matter, yet expressed in extremely accessible language by anyone willing to put in the effort to think it through. You should LOVE it!:-P It's entirely free too, though hopefully people donate to help him make more of such fantastic material.
Anyway, imagine if you will: a bird does not "know" how to build a nest, they just do it. If asked, a bird cannot explain the matters of structural integrity, materials resiliency, and so on, even if it somehow could speak (or like if you could read its thoughts). Even so, it manages to accomplish the task b/c of the instincts built into it - i.e., as Daniel Dennett explains (famous atheist apologetics philosopher, here I refer to his book Darwin's Dangerous Idea, though the following is my paraphrase not direct quote), "Nobody is arguing that design work was not done, the question lies rather in who did that design work - a God or the blind natural forces of millions of years of evolution."
In short: conservatives can be quite effective, even without having the slightest clue of what precisely they are doing or why, as in they (or imagine another hypothetical example of a con-man or -woman) may not know the details of the underlying processes by which it works, yet they still recognize what works vs. does not, and act to exploit what they see. Those January 6th protesters, despite being traitors who tried to perform literally treasonous and fully murderous acts within Congress and the White House, at the same time also thought of themselves as "patriots" who were there to "defend the Constitution" - despite most never having read the actual primary document, or how incongruous those two thoughts would seem to anyone who gives the matter even 10 seconds of rational thought, yet despite that they literally believe both at the same time, and others who were not there choose to believe that it was "peaceful". The Alt-Right Playbook series explains how this can be accomplished, but even from what I have said so far, you can see that it works.
Another quick peek into it is that few people believe all the crazy stuff, nor is the goal (of those putting it out in the first place, e.g. many of the material has been directly traced to Russian interference e.g. anti-vax memes) even to get someone to truly believe even a single one of them. Rather, the goal is to foment distrust in the overall systems. Which let us be frank: the government truly is corrupt - the conservatives do not have to lie about that much at least - and so then that creates an opening with which they can further widen the divide, maybe even getting them to do something more drastic like secretly tamper with COVID vaccine stocks even in the midst of the pandemic. And then in the meantime, they vote Republican, which has the effect of halting aid to Ukraine, which furthers the Russian agenda - so despite the fact that whatever happens to us here in the USA may even be mostly irrelevant, still we (the general populace) have been used to further an agenda that is not our own. Yet, as time marches on, it might become so, and we may one day form an alliance together with Russia, rather than act to repel it, if Trump gets elected again. Russia is using a quite brilliant strategic maneuver there - what boggles my mind is that it is working. But, we (Americans) are weak as we have faced no true aggressors in most of our lifetimes - even Boomers grew up after WWII, and Korea and Vietnam were somewhat far-off affairs, as too was the Gulf War. We could not imagine a nation (Russia) wanting to literally eat us alive. So we seem to have not truly feared its influence over us anywhere close to the degree that we should have.
And that video series explains in large part why: b/c of the onion-layer effect whereby people get sucked in by the more inoffensive material, even while being groomed to be brought deeper in to the more insidious stuff later on, but only when they are ready do they finally start to realize that all their "joking" previously (like: "it's the fault of the Jews!:-P", which they did not truly believe... at the time) has, at first, a grain of truth inside of it, and then later, that it was never a joke to begin with at all. However, if they were to have been presented with the full Nazi/fascist agenda all at once, they would have turned it away, yet like a frog in a cooked pot (a false analogy btw - frogs are smart enough to jump out, it is people who are that dumb as to not!!) they continue to be changed by marinating in the shallower stuff, until as they get deeper and deeper over time, one day they finally are ready to accept the whole reality. Or not, but even so, them remaining in the shallower regions still helps further the agenda overall - e.g. in voting, and in being able to use those people like a magic story's mage or a necromancer uses a "meat shield": to deflect attention away from those who truly do believe.
Getting back to what you said: does Musk, or Trump, truly believe in what they are saying and doing? I would rather rephrase the question to be instead: does the authenticity of their belief systems even matter, when they are acting effectively in the services of the facists either way? At what point do they need to cross over to become a "true believer", when all along their actions were operationally indistinguishable from one anyway? Then, as you say there are people such as Greene, who regardless of their "true beliefs", are nonetheless useful to the cause.
I applaud your asking these questions - so few people do it seems. True answers are difficult to come by, requiring many hours (and days/weeks/months, maybe years?) of thought, though fortunately there are resources that can help speed that along, and yet how does one even find such things in this era of enshittification where you cannot trust half 90% of what you see & hear - but even so, I maintain that some things like that video series have the "ring of truth" about them, and that once you see the logic behind it, you can never go back to not knowing ever again. Enjoy it!:-)