A conspiracy theory proven true is no longer a conspiracy theory, it is a boring fact. Conspiracy theorists aren't going out on the streets protesting the gradual erosion of civil liberties. They're calling that a limited hangout and demanding the real juicy shit, even if it doesn't exist.
Abuses of government power are generally boring facts, like civil asset forfeiture. “A conspiracy theory is an explanation for an event or situation that asserts the existence of a conspiracy by powerful and sinister groups, often political in motivation, when other explanations are more probable.”https://en.wikipedia.org/wiki/Conspiracy_theory
So, when cops rob an armored car via civil asset forfeiture it’s just something that happened no conspiracy theory required because there isn’t some other example that fits. But, the moon landing was fake fits because there’s another explanation.
Occasionally what once was a conspiracy theory is now considered factual, but at that moment it stops being a conspiracy theory because it’s no longer fringe. As such this isn’t a pejorative definition as it doesn’t directly imply such theories are incorrect.
In many cases, yes. You'll often find the actual conspiracy that turned out to be true only had a tangential relation to (and few if any specific details in common with) what conspiracy theorists were talking about. Although the conspiracy theorists will always take credit for "being right all along."
And invariably, the actual conspiracy gets revealed by parties other than conspiracy theorists themselves, because the conspiracy theorists don't actually have insider knowledge, they're just doing the strings and thumbtacks thing and making guesses.
In my experience, conspiracy theorists don't have a lot of interest in these. Unprovable conspiracy theories are more interesting because it affirms their sense of paranoia, identity, feeling special, entitlement to the truth.
Conspiracy theorists might go on and on about JFK or 9/11 or Pizzagate, but how often do you hear the conspiracy type obsess about Jan. 6, an actual proven conspiracy to overthrow the government of the United States? The "proven" conspiracy theories they do care about, like MKUltra, are generally cast as far more consequential than most people would say they actually are.
That is technically an example, but I think it also proves OP's point. Faced with a clear, public, and well-understood conspiracy, the theorists can't accept the demonstrated facts. For the surface truth to be the actual truth is intensely unsatisfying and disempowering. There must be something more and deeper to it, always.