I watched this recent video by Dogen about how immigrants should adapt to Japanese life etc.:
https://www.youtube.com/watch?v=9_tXp5sFlHQ
One of the things I found odd was that he mentioned "eating while walking" as one of the offensive behaviors foreigners should stop doing. Many of the other examples made sense to me, but for this one - I never thought this was "rude", rather "odd". Thinking about it, I had a few conversations about this over time, but never got negative vibes for it (at least directly). Might be that people don't want to tell me it's rude, but I also got no negative opinions about it from people who lived abroad for a while (and thus are maybe better at communicating with foreigners) and/or are usually more upfront with me.
So my current understanding: It's odd but not rude. Thankful for any further insights.
I really can't think of any food that I can eat while walking. The food I list would just invite disaster on my clothes if I eat them while walking. Maybe candy bars or small snacks.