Oh, they don't all have to; it just needs to be permitted -- which it is.
All you need is one Ship, remember. One.
Yes, nobody "has" to do what the Minds say but in the end, they do when the Minds want, don't they? Because the Minds are so much smarter than everyone else that they can manipulate anyone to doing their will. Don't believe me? Ask Yime Nsokyi.
So yeah, a society in which a class of supremely intelligent machines can control -- either directly or not -- every other sentient being they can reach is not one in which I'd say the organics have much autonomy. Sure they can do what they want but why bother doing anything at all -- the Machines have it well in hand.
They're meatpuppets: toys and pets that exist on the sufferance of the Minds. The fact that the Minds are fond of their pets does not change this essential fact.
EDIT: As far as Corrupting the Minds works I am assuming that they have some sort of presence or method of communication with the Physical World. Through that presence they can receive information and their instruments can become "blessed" by their new Patrons. The Chaos Gods do not need to physically "touch" someone to corrupt them -- except for Nurgle who is a really touchy-feely kind of God.