Yes, among the many things wrong with schools, I would have to agree with you that one of the biggest is what kids can do with free time.  I know kids like your daughter.  They understand what is going on, get to work, get it done only to be rewarded with "Now sit down and shut up."  I am revulsed by this to the degree that I started a soccer league for the kids who have to sit down and shut up waiting for the bus in the afternoon.  Now, it is not the fault of the system that of all the people watching those kids during that time, only one is willing to take em outside and let them run.  At the same time, it is not practical to let kids move about freely here.  There are just too many kids here.  While kids are curious, they are also not that motivated.  I congratulate you on having an exceptional kid.  

I do know the model for our public schools.  I spend some 50 hours a week here and it is DEFINITELY NOT democratic.  I work for an authoritarian, Machiavellian principal. The downside is as a teacher I have to waste my time "keeping up appearances."  When that door closes and those kids are in the room, it's my business.  I do not run my classroom in a democratic manner.  We have goals to reach, and they are not unreasonable for most.  When the work gets done, we decide how to spend our free time as a group.