I think that the hidden question here is, does writing require teachable skills. Of course writing requires skills. You have to be able to make marks on paper with a stick, etc. The real question seems to be, does it require skills you can't just pick up by living life. Do you have to specifically study writing skills in order to be a writer?
This is really a question about what type of skill writing is.
Is it a skill like walking, for instance? We don't teach our kids to walk. (They don't have enough language to take instruction anyway.) Kids teach themselves to walk. They learn by trying to improve their ability to get from A to B. Some of the intermediate means they discover in the process are pretty funny, before they finally figure out walking. Walking is learnable, but not teachable.
Skills like this are tacit skills. Even after we learn them, we cannot fully express how we do them in words. The body has learned, but not the part of the body that knows how to translate what it knows into words.
Then there are skills like how to ride a bicycle. We teach our kids to ride bikes. We can write down a pretty good description of how to ride a bike. Still, there are parts of bike riding that are still tacit. Even with the best instruction, you still need to fall off a few time before you get comfortable and secure. Bike riding if partly teachable. But people can also figure it out for themselves. It is wholly learnable.
Then there is installing a home theatre system. This is a fully teachable skill. A manual can tell you how to attach all the wires in the right places and if you pay attention, and the manual is correct, you can complete the task successfully the first time. This is a fully teachable task but is not really learnable. You need the information in the manual.
So what about writing. I think we should class it as fully learnable and partially teachable. To say that it is only partially teachable is to say that it is partly tacit. There is part of writing -- or at least of writing well -- that we don't know how to express or explain in words. (Ironic? Sure. But remember that language itself is like this. We don't have a complete understanding of how it works.)
Why is it fully learnable? Because writing is all surface. You can see 100% of what the writer has done. Every part of the technique is there on the page to be learned, even if we can't put what we are learning into words (just like walking). In this sense, it does not require any teachable skills. You could learn it all tacitly.
But, like riding a bike, a good deal of it is also teachable. When we teach skills we break them down and give them names and you can then list specific skills you can learn. In this sense a "skill" is an artefact, something we have captured and defined so we can teach it.
Since writing can be learned entirely tacitly, it does not require any "skills" in the sense of these captured and defined teachable skills. It can be done based entirely on tacit skills, at least in principle.
That said, in practice you are likely to make much more progress if you learn the genuine teachable skills. The trick is to tell the true teachable skill from all the silly rules out there masquerading as writing skills. And you also do have to remember that writing is not wholly teachable. There are tacit elements to it that you only learn by reading and writing.