I think a lot of this simply depends on the distance between home and school (or other places kids need to go to) – so it's the difference between a compact city and sprawling suburbs.
Where I live (central Europe), the density of public elementary schools in cities is high, so kids walk there alone. The density of secondary schools is lower, so most kids use bicycles or trams / buses.
Interestingly, there are a few private elementary schools (usually english speaking) for children of expats, where cars queue up in the morning, while parents drop off their children. I've never seen this at public schools. I believe this is because there are only a handful of those schools, so they are further apart – and maybe also because the parents (growing up in the US or UK) are already conditioned that this is a normal thing to do.