At least where it is a paid staff member who is going to the school anyway, they will be reliable. An ad hoc arrangement between various parents is open to let-downs, misunderstandings, "something cropped up", potentially leaving small children waiting unsupervised or at their gate. People are generally more selfish these days anyway, and unless someone is "officially" tasked with doing their turn then it could all go wrong. If the walking parent's own child wakes up ill, are they going to leave him alone in order to take the others to school?
How does the walking bus work? Presumably the children will have to get to a central location before joining it, as it would add miles to their walk if they have to meander down side streets to collect everyone from their homes.
Funnily enough, when we last stayed overnight in York, in a little hotel about a mile from the station, the next morning we went for a walk and saw a group of probably about 20 uniformed primary school children marching along, in the care of one adult male following behind. It was a quiet road so he could watch them all, but I did wonder what would happen if the convoy went into a busier street. I'm not sure what the ratio should be, but going by the school parties I've seen on trains I'd think about one adult for 6-8 kids, possibly more if they are very young.