(More liberal) catholic speaking here, I think it's because those theories are outside of the bounds of catholicism and have little to do with the visions and Fatima. I'd say 'extra-catholic conspiracy' instead of 'intra-catholic conspiracy and intrigue'
So if you are an orthodox (in the sense of traditional) catholic you can acknowledge that pope Francis is a legitimate pope, but you can have legitimate criticism (as a catholic you are to follow both your own conscience and church teaching). Criticism like perhaps you think pope Francis is deluded in some areas of the Faith (some controversial lines in Amoris Laetitia for example). Catholicism allows the possibility, it's all part of God's plan. A common misunderstanding is that the pope is always infallible The papal infallibility dogma has been. written down in 1870), and means he's only infallible when he is speaking or writing 'ex cathedra', which means:

"when, in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, [the Bishop of Rome] defines a doctrine concerning faith or morals to be held by the whole Church."
https://en.wikipedia.org/wiki/Papal_infallibility

Something akin to saying that the specification of the Faith has been clarified or updated by the pope in lieu of Catholic theological consensus or lack of it. For example the TLS spec update stating compression isn't kosher from now on.

1P5 (onepeterfive.com) is an example of such a stance or Fr. Z's blog (wdtprs.com) - but these authors are still inside the bounds of regular catholicism and do not entertain irrational views like "pope Francis is an imposter pope" as Benedict XVI clearly acknowledges the papacy of pope Francis.

The particular 'conspiracy' which is talked about in Slate is the content and meaning of the so called secrets of Fatima themselves where traditionalists might have some supposedly visionary edge over modern catholic theology and a suspicion that the Vatican is not telling the whole story for some reason, this leaves room for legitmate speculation or some flight of fancy.
There are of course numerous other conspiracies surrounding Fatima or the Catholic Church, but that doesn't make them necessarily catholic, catholicism is also about using reason.
https://www.philstar.com/opinion/2011/04/01/671497/there-4th...

1P5 demands revealing the complete 3rd secret: https://onepeterfive.com/third-secret-fatima-kiev/

Fr. Z says the consecration to Mary can't hurt: https://wdtprs.com/2022/03/25-march-francis-in-rome-a-cardin...
and speculated on the third secret earlier:
https://wdtprs.com/2020/01/113187

The traditionalist bishop Schneider and opponent of pope Francis on the consecration to Mary today:
https://onepeterfive.com/bishop-schneider-pope-francis-conse...

https://en.wikipedia.org/wiki/Athanasius_Schneider

Hendro Munsterman, a more liberal theologian on why the consecration to Mary is not a good thing: https://threadreaderapp.com/thread/1506961091229888516.html
He also wrote this article on the Russian Orthodox/Ukrianian Orthodox/Ukrianian Greek Catholic/Roman Catholic divide:
https://www.areweeurope.com/stories/understanding-the-russia...

A rather traditional Canadian catholic priest about the status of catholic prophecies:
https://www.youtube.com/watch?v=bP38Kw66_bo