I think strong writing skills are absolutely an advantage for anyone trying to build a website, blog, or serious online presence simply because the written word is the dominant method of communication online. People are more likely to recommend and return to a site that contains readable, quality content.
Of course, for those with a good idea who lack writing skills there is no shortage of content suppliers these days where fledgling professional writers will write good copy for a minimal cost, and many profitable web sites use ghostwritten material.
If you are a good writer you don't have to outsource your articles which will save you a lot of money in the beggining when you don't earn from your sites yet. Personally, I only outsource articles about niches I know nothing about. That being said I write about 95% of my content
My main profession of website development doesnt require me to be a great content writer. That said, when you are from a non-English background and catering to the people where English is native, you do need to have skills of correct language - so that you can communicate better. My ability to understand and speak better than many of my local guys has helped me a lot.
Though I also do write content for a few clients, I am not much into it these days. But yes, if this is what you intend to do, you should have good vocabulary and express through writing powers.
Being a writer may be an advantage but not always. If you are building a website with funny pictures, your writing skills won't help you much. For content websites, writing skills are a necessity unless you plan to outsource the entire writing work.
But personally I believe that to be a good company about web design building, you should be a good manager on top of all. Like that you gather a good team with designers, programmers, content writers etc.
You don't have to be a good writer if you have the money to hire good writers or if you don't have too much writing on your websites. In the beginning, most people need to write their own stuff unless they have a game site, a video site or something on those lines. So, it all comes down to the type of sites you are building and the money you have. Good writers are present on freelance websites and can be hired to do all the writing you need.
Like Barry.Lejieg said you need to be a good manager that knows what should go where. If you can put the proper people in their most suited places, your site will be a success whether or not you write well.