There is something, almost fundamental in the nature of conspiracy peddlers, that makes them try to link anything to make their conspiracy more plausible, even the most tenuous of links. Sadly, they don't have to try and completely prove anything - if it sounds plausible enough, people will assume it to be true. So, the problem is not with the originators of conspiracies, a lot of the problem comes from those willing to accept it at face value.
On the subject of the Illuminati, there once was a public organisation called the Bavarian Illuminati, and I believe they operated around the time of the signing of the Declaration of Independence in the US. So the conspiracy theory goes, the influence of this organisation continued long after it was outlawed, and survives to this day, as a sort of Machiavellian 'power behind the throne' organisation. Whether this is true or not is probably impossible to prove, and almost certainly not the issue. The issue is in the continuing need to believe there is something more that is going on behind the scenes of power.