I came here to answer this question again because I feel like I omitted important points.
Schools don't present your typical production function. They're multi-product firms with several inputs with varying degrees of substitutability and complementarity. Different outputs (such as human capital, student health and socio-emotional skills) aren't necessarily antagonistic, but compete for school resources and there likely are multiple equilibria, given the vast amount of inputs and multiple outputs.
One important feature of a school's production function is that customers are a central aspect of the firm's technology. Students aren't just customers, they're inputs and each one has an unique, time-varying marginal productivity. This implies private schools should charge students different tuitions, in some cases provide full-scholarships or even pay students to attend the school. A good student, through peer effects, can alleviate the school's input demand - especially for teachers and TAs.
So, yeah, you're right. Usual pricing strategies used in "traditional producer theory" aren't exactly appropriate for studying the "industrial organization of schools". Because of (i) the variety of input substitutability and complementarity, (ii) customer-input technology and (iii) the presence of various outputs.
Some schools, for example, focus on the niche market of "conscious" parents and devote more resources to socio-emotional skills. Others go for the classic "we educate geniouses here".
Winston (1996) is a good reference on this topic.
I hope I made things a bit clearer.