Yes, I agree - however, the fact that ALL citizens have a say is not irrelevant.
The case you're talking about is one where the State acts in its capacity as an agent of the citizenry, similar to how a manager can hire, fire or sign contracts within the bounds set by the owner or owners. It doesn't mean the manager owns any property, simply that he as an agent of ownership is currently in possession of it.
In the capitalist public/private sense, yes; in the socialist personal/private/social sense, this is social not private property (specifically, cooperative social property.)
The sense of “private property” opposed by socialism is not at all the sense which is opposed to “public property”.
It is entirely possible for people to create a super-capitalist, an image reflecting bourgeois nature, within a cooperative establishment.