The thing is that “a good writer” is not necessarily a писатель (i.e. an author who writes fiction). In fact, it can be any person who writes. So “he is such a good writer” expresses the same idea as “he writes so well!”, but sounds better and is used more often by native English speakers.
Expresses the same idea, yes, but it isn't the same translation. Here, the word пишет is clearly a verb, and needs to have adverbs modify it. I think native English speakers will be more confused if they think "he is such a good writer" and "he writes so well" are equivalent, because then they cannot understand why we must use так and хорошо but can't use такой and хороший. These phrases are idiomatically equivalent, but they are not grammatically the same.
Btw, I am a native AM English speaker.
In professional translation, using the most natural phrasing IS most important. But when learning, one must first understand basic grammar before taking the next step to natural interpretation in the target language. A student who is learning the difference between a how a sentence is constructed using a verb vs how one is constructed using a noun is better served by learning correct grammar first. When I first saw this sentence I was confused; why not такой/такая/такое like in Твоя музыка такая громкая? Then I realized так was an adjective in that sentence but an adverb here. If I had been told that Он так хорошо пишет also translates to "He is such a good writer" I would still be confused. When I realized так and хорошо modified a verb, пишет, everything became clear for me.
As for your question, it depends upon context. If I am speaking about a writer who can move me emotionally with a single paragraph, or about a poet, "He writes so well" would be my choice. If he writes a lot and sells many books, I would use, "He is such a good writer" and if he is a student learning to write by hand and doing it well, I would say, "He is so good at writing". All three basically mean the same thing, but notice that "write" is a verb in two of those sentences and a noun in the third, and the word changes form in all three. Gramatically, they are not the same. A student learning English must understand these differences if they ever hope to grasp English.
@NovemberVee: I’m glad to know you’re a native speaker; as a native you know better. If you were to choose between these three sentences: “He writes so well!”, “He is such a good writer!” and “He is so good at writing!”, which one would you rather use? As a translator, I am sure that, in translation, getting the meaning across in the most natural form is much more important than keeping the original syntax structure. But I may have made a mistake when I suggested that one structure is used more frequently than the other. I should’ve added “to my knowledge”.
The word хорошо in the audio is pronounced as if it was a question. To make the sentence sound as an exclamation you should raise your pitch from он to так, then make it fall either all the way through the rest of the sentence or at least as far as the end of the word хорошо. In the latter case, you would say пи- in пишет at a very high pitch and pronounce -шет at a mid-low pitch.