If I have a google sitemap, but it's not displayed on the home page, will it be disadvantage or even no use in terms of SEO?
I think by google sitemap, you are referring to sitemap.xml. It would be a good idea to have a link from the homepage but if you don't have one at the index page, don't worry. But remember to submit the sitemap.xml to google through google webmaster tools whenever you update your contents.
I don't think sitemap.xml should be visible on the home page or any other internal page of the website as this sitemap is basically used to navigate the SE crawlers. To navigate the users on your site , you can add a simple sitemap containing all the links url on your home page...