At least in the US, public schooling initially was based in reformist desires to enlighten the masses, not as government control. Whether they are controlled by the government to an extent now - I think that is open to interpretation.
There are a number of options to public schooling, like religious schools, magnet schools, global schools, immersion schools, and so on. A number of my peers in high school got certificates in nursing, auto mechanics, and bricklaying.
It is true that there are many improvements that can be made to schooling, but having worked in a public school, I can also attest that a huge problem is that many parents and students don't have any interest in learning, period. By middle school, a depressing amount of parents are just looking for free babysitting and students just want to spend time looking at reddit or porn.
How can personality psych be used to improve education? Probably along with learning about multiple learning styles. Introducing it early, getting teachers on board with teaching in numerous different ways to reach all sorts of different students. Encouraging lots of variation and "tracks". Not trying to send everyone to Harvard and acting disappointed if they're a skateboard pro instead. Trying to help each student shine of their own merit. Introducing real-world classes like practical money and finances so students can make a life of their own successfully instead of being tracked into academia without ever learning how to do their laundry.