From my experience - another reason why grass(ed) tracks work well is that it provides a visual barrier for pedestrians, encouraging them to use crossings. The grass tracks are clearly not the road, nor is it the sidewalk. However you could still dash across if you really wanted to.
Putting up pedestrian barriers in the name of safety has been shown do to have the opposite effect - drivers feel "segregated" from other road users and then speed more, increasing crashes! Consequently, many cities across the world have been removing them...