I have no idea if his Prussian school system claim is right, or whether the idea that this was all intentional is just a matter of seeing agency where there is none due to bias.

It's at least partly right, but strikes me as weak evidence. Horace Mann and several of his contemporaries admired the Prussian system and introduced reforms to American schooling based on it -- but he probably wasn't trying to foster exactly the same features that later commentators have objected to. There have also been several major changes to American public schooling since then, many of them divergent with the development of the Prussian (and, later, German) school system.

It's plausible to me that the American school system functionally prioritizes inculcating obedience to authority, whether it was designed that way or fell into that arrangement through a process of evolution. But that's a claim that's got to stand or fall on an analysis of the system as it currently exists, not of its remote origins.