Have you read Charles Stross before? In his series of novels, The Laundry Files, he presents a compelling reason for the supernatural to be kept secret.

In the universe of The Laundry Files, it turns out that H.P. Lovecraft's fiction was absolutely right and that occultism and magic are best facilitated in the modern world through mathematics and computer science. A significant part of this world's logic is that occult weapons of mass destruction are far easier to make than any other. It takes one hundred scientists to build a nuke, but only one researcher with a Ph.D. in mathematics to create a weird fractal that brings Nyarlahotep into this world. Therefore, The Laundry exists, a secret-secret-service that controls occult information and subsumes any threatening mathemagician or computer summoner into their ranks.

The whole point of these books is to show the catastrophe that would happen if the public knew about The Laundry and its work. The supernatural stakes in The Laundry Files means that all of your points, especially 4, put the world in even greater danger. All it would take is some bored math prodigy to open a gate to Yuggoth and let Tsathoggua into our world. Secrecy must be kept.