Which is completely fine. If Google did not support managing websites through a URL, why would they possibly offer this method in the first place?
Adding your website to Search Console helps to get your website in the search results, but doesn’t automatically make you the top ranking site for any relevant search query. You’re still reliant on Google to crawl your website and rank it in the results. I’ve heard it can take weeks for Google to crawl a new an unknown website.
When checking which pages have been indexes, try doing a search query like
site:example.com with your domain. If you just search for the name of your website without explicitly filtering for your domain name, your website might just be ranked at the bottom of the results. And no hosting service in the world can guarantee you that can’t happen.