This sounds at best correlation not causation (and the correlation is pretty weak too). I didn't see any causal effect.
My instinct tells me that walkability is a consequence of the actual cause, which is probably higher property values in denser cities, meaning better schools, or something to that effect.
I was living for some time in a city that offered a lot of options for pedestrians but certain things were not available or only with difficulties - e.g. buying furniture - unless you had a car. So living there without a car was fun at first but then it became very draining, especially when having to move from one apartment to the next one.
The logical explanation is probably that with a car one has not only more options but also options are available at lower cost. E.g. one can move to a cheaper area, with cheaper supermarkets but that is only reasonably accessible by car. The irony for me is that cars became that popular in the first place, given that they are a quite unreliable and expensive toy.
My instinct tells me that walkability is a consequence of the actual cause, which is probably higher property values in denser cities, meaning better schools, or something to that effect.