I think the key point here (which the author kind of dances around) is that walking puts you in a different kind of mental mode. He writes, "walking makes me smarter".

This jibes with my experiences when I started walking 5+ miles a day (that was before I had young kids that absorbed every minute of free time). I felt like it did make me smarter. Or at least, better at programming and writing.

I now commute by bicycle, and I think that is somewhere on the spectrum between driving and walking. But even on a bicycle, you have to be in "collision avoidance mode" pretty much continuously. Your mind still wanders -- but with a lot more interruptions and alerts.

I think there is something "magic" about walking -- by which I mean something great, but something that we may not be able to scientifically understand in my lifetime. We know we evolved to walk upright, and we can measure a lot of the physical health benefits it confers, but we still don't know enough about the brain to prove that it benefits cognition, happiness, satisfaction, etc.

I strongly suspect that it does, though.