Its less to do with cars and more to do with risk aversion imo, I walked to and from Balgreen Primary and Tynecastle High from the start and on my own from about (P3) onwards both these schools from my house required negotiating Gorgie road which is much quieter than it used to be.
My mum was less worried about Gorgie road than she was the rat run round Robs Loan in Chesser, once that was blocked up i was free to go.