Wouldn't accepting 1000 applications, and teaching Montessori-style to 500 of them and traditional-style to the other 500 pretty accurately control for all of those self-selection biases?
(I walk pass 2 schools on my way to work each day - one Montessori, one not, sometimes around the time where children are dropped off by their parents. Even not knowing anything about the relative merits of the systems, I can easily tell which school I'd want to send my daughter to in a few years time; at the risk of sounding elitist, it's like standing next to the exit of an opera performance vs a soccer stadium).