It doesn't have to be, does it?

Actually, with anything close to the current set-up, I would say yes. Or, maybe I'd have to flip that around a bit to say that the current set-up works best for indoctrination. Which makes sense. The bare basics of how public (and most private schools) work goes back over 100 years, which is a huge problem in and of itself. One of the earliest countries to institute public education was Bismarck's newly formed Germany in the 1860s and 1870s. The express purpose was indoctrination, to help create a single national German identity out of the myriad of small countries absorbed by Prussia in the unification. Other countries, including the US, followed. The system started for indoctrination, and has a very nice by product of creating a more productive nation as well.

We can also look at some specific policies that would need to be completely reversed.

Truancy makes no sense if public education were really for education. Being in a classroom doesn't make you learn. In fact, having kids who don't want to be there in a classroom often ends up hindering the learning of those who do want to be there. What making kids show up does work for is indoctrination, as they then must interact with others and are exposed to the ideas, even if they don't learn it.

Other parts of the system, like required classes, grades, and standardized testing plays into it too. By requiring certain classes, especially in the humanities, schools help to expose students to ways of thought and ideas that are in line with society, and creates a common language for citizens to work with. By tying in grades to future success, the education system makes working within the system absolutely essential. Standardized testing works the same, but at much higher stakes.

For all of that, I don't think the indoctrination is all that bad. It isn't necessarily indoctrination into a specific way of thinking (or at least, in many schools, and ideally, it isn't). The indoctrination is in how to function in society. I have some cousins who are homeschooled, and interacting with them can be frustrating and tiring because they just miss things like social cues, or manners, or standard forms of interaction because they don't interact with a large number of people daily. Seeing kids transition from being homeschooled to attending public school can be interesting as well, because they just don't really get how wider society works. They also tend to be less independent from their parents, which creates other problems in interactions. kleinbl00, I gotta warn you too, private school kids can be almost as bad. Private schools can be very homogenous, and so I've seen a lot of kids struggle with having to interact with people who they don't have as many shared experiences with—kids from different social classes, or races, or even just different parts of the city. So, since you're set on private school for your daughter, make sure she has some other social outlets, like a sports team or church group or Girl Scouts or summer camp or debate team or something. The kids I know who have transitioned best from homeschool or private school to public school, or even just to society at large, have been those who gained part of the social indoctrination they missed out on from other activities.