In your particular example, only 我快速跑向车站 is used and can be understood without some spiritual raising of eyebrows.
Before everything: I want to comment on 快速地，我跑向车站 vs Quickly, I ran to the station separately.
Back from digression, in order to generalize what we see in 我快速跑向车站, first, I would observe that modifiers of verb, noun, adjective, are always immediately prepositioned in modern Chinese. Perhaps that's also true for all verb phrases and noun phrases -- however one may discover the plausible phrase structures in Chinese.
See this primer on 副词 for some examples of modifiers (副词 is modifiers for verbs and adjectives). You can discard the claim of the positioning at first and just borrow their examples of 副词; then consider other 副词 they may remind you of. Can you think of examples of post-positioning? I bet it is difficult!
Next, I would observe that 快速 is not a valid modifier for anything in your example sentence 我跑向车站 except either 跑 or 跑向车站.
I am less willing to conclude that there is strict syntactical structure aside from the immediate prepositioning rule for modifiers. But common Chinese sentences are heavily guided what would commonly be semantically consistent. If it is not semantically valid, it will raise spiritual eyebrows reflexively.
That is how I would theorize why only 我快速跑向车站 is grammatical.
As for "快速地，我跑向车站" vs "Quickly, I ran to the station".
When you read "Quickly, I ran to the station", do you really see "quickly" as being associated to just the verb "ran" -- especially when you first read/hear it? Or do you absorb the idea of "quickly" and then allow it to be added to whatever follows would convey?
I would say in any language, you may attempt to extract a modifier phrase (ie. adjective phrase, adverb phrase, preposition phrase) to the front or rear of a sentence and there is a chance that, even though not common used, it is tolerated and even used somewhere, perhaps in poetry. Unless a sentence structure modifier phrase + comma + "verb phrase with noun" is commonplace in a language, I would hesitate to consider it a part of the grammatical system.
The core idea in my answer is that (1) I suggested that there an "immediate prepositioning rule" for modifiers in Chinese. 我快速跑向车站 is an example of that.
But there are some more nuances which may be of interest to you. Listed with (2)-(4) below to facilitate quick reading as this answer is getting very long!
(2) Chinese modifiers often are not morphologically different when applied to noun, adjective, and adverb. As a result, there is a sense that you can flexibly try placing the modifier 快速 in most places in the sentence. However, none can semantically work.
(3) I would suggest that there is an emphasis in the customary association of words that would be semantically sensible to combine. Hence, in the example of 我快速跑向车站, if you try any other order, it would perhaps raise extra spiritual eyebrows, giving an illusion of inflexibility.
Finally, let me touch on the immediacy part. With what we said above, do we really have an example of non-immediacy for adverb in the English examples?
When taken the view of adverbs may modify verb phrases in all of:
- I quickly ran to the station.
- I ran quickly to the station.
- I ran to the station quickly.
"quickly" is either immediate before or after a verb or a verb phrase. So if you consider the adverb as modifying either verb or verb phrase, (4) you never lost immediacy.
I would be intrigued to find examples of non immediate modifiers being commonplace in any language and if someone can comment the examples they know, I would love to be intrigued!