This is a response to a video, but I can't for the life of me remember who it was I responded to. Their point was that conspiracies are difficult because people are dumb.
The argument that secret societies can't keep secret for a long period of time because "people are dumb" doesn't follow. People are smart enough to build bridges, conquer empires, get elected, build huge businesses, repair cities after devastating earthquakes and many other things. Are we supposed to believe that keeping a secret is harder than all those things?
In fact your own expample of the Ministry of Magic shows that, where it concerns their interests, people are very smart. Consider how much work the MoM put into convincing everyone it wasn't a totally useless, largely corrupt, incompetent mess of an organization. It worked. They managed to do that with very few exceptions and those were mostly people who had special knowledge of MoM screwups. This is an amazing accomplishment that is almost (?) as hard as doing their ostensible jobs. So you have an example of a successful conspiracy. You don't need perfect people to have a conspiracy/secret society you just need the right incentives.
If it's hard to keep secrets then why did the mafia last for centuries, and stay so secret that the government denied their existance for decades? Other conspiracies that worked for a long time were the British governments secret attempts to get the USA into WWII, which wasn't exposed until people started writing memiors, and the Conintpro FBI programs. Keeping secrets doesn't require a lot of intelligence it just requires that you only communicate with people who will keep the secret. If you have a system where there are a lot of people who benefit from keeping the secret or would be harmed if they revealed it, such as an intelligence agency or corporate beuracracy, then just having them talk mostly to each other works.