Why cannot verbs like 歩く、走る、泳ぐ be used with に or へ?

Question

I've learnd that: verbs like 歩く、走る、泳ぐ cannot be used with に or へ, and only まで can be uesd in this case. For example:

時間がなかったので、駅 (〇まで/ Xに/Xへ) 走った。

But when it comes to verbs like 行く、来る、着く，both に,へ and まで can be uesd. For example: 時間がなかったので、駅 (まで/に/へ) 自転車で行った。

But I cannot figure out what's the standard of judgement in the permission of the usage of に and へ. At first I thought it might be "whether the verb expresses movement or not", but since 歩く、走る、泳ぐ and 行く、来る、着く can all express movement, then what's the standard of judgement to distinguish these verbs into two different categories, in which the one allows the usage of に or へ, but another doesn't allow?

I still have some confused points after reading this:

1. The standard to distinguish"pure motion verbs" and "motion manner verbs" seems unclear to me, since 動く and 移動する also don't necessarily need a direction(e.g., 一歩も動けなかった,遊牧民たちは絶えず移動した).
2. Why can まで be used in "motion manner verbs", where the usage of に and へ are ungrammatical?

Answer 1

Verbs like 歩く do not have a specific direction. While に by itself doesn't bring directional meaning either, it becomes so only with directional verbs like 行く. This is why you either need to use other directional particles like へ (towards) or まで (extend, up to a certain point), or use compounds like 歩いていく.

I can try to explain it like this. When we talk about movement verbs, it's very common to have 2 different meanings. One meaning describes how action is done, the manner or activity, another described a location change from A to B.

Walking improves my mood. This is why I try to walk every day.

I walk to the shop.

The former doesn't have any natural end point, unless we restrict it with a goal. Activities can be ceased, but not finished in the same way as action like "build the house". Can we say when walking actually ends? In 20 minutes? In 2 hours? Person just gets tired and decides to do something else. In English both usages more or less possible. I feel a bit unnatural about using "to walk" to simply mean from A to B, and I think it's because it rather describes the manner, than the location change, but it's still acceptable. In Japanese such shift is stronger and 歩く by itself isn't enough to get directional meaning.