> A democratically run company is still an owner of private property,
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”.
When I say "company" I mean such an organisation as which employs wage labour to further the accumulation of capital; such an organisation could not exist under a Socialist mode of production, at least in my understanding. This is because it is firstly not wholly the "property" of the workers in general, because accumulation of capital requires exploitation and value added through M-C-M'; a cooperative still sells produce, so in my view this makes it inadequate for Socialism. Perhaps something like mutualism would better describe such an arrangement at the scale of a society.
It is entirely possible for people to create a super-capitalist, an image reflecting bourgeois nature, within a cooperative establishment.
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”.