The person who wrote this wanted to say "I will walk to town".
According to a classic explanation, "pure motion verbs" such as 行く, 来る, 動く, 入る and 移動する can take both ～に and ～へ, but "motion manner verbs" such as 走る, 泳ぐ, 歩く and 飛ぶ tend not to take ～に. Verbs in the latter group do not even need a destination (e.g., 部屋をうろうろと歩く, プールで泳ぐ, 空を飛ぶ).
That said, there are many cases where ～に is used with 歩く/走る/飛ぶ. This is extensively discussed in this article. The author says that when the manner of a motion or a situation is more focused than the motion itself, ～に歩く tends to be more tolerated. ～に走る/～に飛ぶ tends to be tolerated when the cause of the action is unexpected and urgent.
I don't know if it will help, but I'll try to rephrase differently what has already been said in your first link above: In Japanese the sense of moving to or from a place is typycally conveyed by basic verbs like 「行く」or「来る」that therefore require a particle like に or へ, while verbs like 「歩く」、「走る」、「泳ぐ」convey the meaning of "passing through", "going across" and as such require the particle を. Therefore, in Japanese you could correctly say 街を歩く「まちをあるく」, but it will mean "I will walk across the town". So, if you want to convey the meaning of going on foot you will have to say exactly that: 街に歩いていく I will go to town walking .