According to Daniel Waisberg, Search Advocate at Google: "Usually, if you have a relatively small website, and your pages are properly linked, Googlebot can discover your content. So, you don't need to worry about the sitemap. However, if your site meets one of the following criteria, a sitemap might help Google decide what and when to crawl your website. If your site is really large, a sitemap will help Google prioritize the URLs to crawl. If your pages are isolated, or not well linked to each other, a sitemap might help Google find those pages. ... Please, remember, that using a sitemap doesn't guarantee that all your pages will be crawled and indexed. But in most cases, your site will benefit from having a sitemap. And there is no disadvantage for having one."
Allow at least a week after submitting a sitemap or a submit to index request before assuming a problem.
The sitemap does not submit your site to Google. Rather it helps Google to understand which URL to index and which URL not to index. So sitemap will help you to index what you really want Google to index.