Because the profit component makes any system more efficient.

At making a profit.

and at providing the service.

Only if all of the following conditions are met:

1. There are multiple options for the customer.

2. The customer is able to accurately asses the quality of different offerings.

3. The cost to switch providers is negligible.

Otherwise, you will either get a monopoly or a lemon market.

1. at least in the urban areas, there are always just as many schools to choose form as grocery stores (so check)

2. easy, just select schools with good graduation, low crime etc. - just as people are selecting good schools now by relocating to a different(better) neighborhood. (check)

3. people switch schools all the time just as they switch jobs, granted you may have difficulties to do it in mid semester but this is not a serious obstacle. (check)

So, all the conditions are met, can we open the market to competition now?