In the title to your thread, you asked: “How important is the title to a website?” Well, for SEO purposes the title to your website isn’t even recognized by search engines, not important at all really. Search engines index and rank individual web pages, not websites, therefore it is your individual page titles that are important for ranking, not your website title.
Since you can, and should, rank each and every individual page of your website, the keywords that you target for each individual page should be included in the title to maximize ranking power. Quit trying to rank your website and focus on getting the individual pages of your website to rank for keywords that are appropriate for the individual pages. You will have much greater overall success if you approach your SEO in this fashion.