God did have visible organizations in the past. And guess what? They didn't work. Look what the Israelittes did when they were given control. They sacrificed their children to false Gods, they set up legalistic structures eventually leading to the Pharisees, etc... That is why our mediator now is Jesus. No imperfect human today, but a perfect high priest. (Heb 5:6)
We can see what happens when humans try to control things themselves today. Some of the guys on here go from one extreme to the other. They go from a mind control legalistic society to nothing. No oversight. No answering to anyone but themselves. They basically turn atheistic or agnostic and reject God altogether because of the actions of 1 organization.
So, does God have an organization? Yes, but it is heavenly. His only representatives on earth are all Christians that spread the name of Jesus.