The best way to make sure you have the right permissions is to delete existing permissions and add the desired permissions you want.
So in your case:
1) Go to the Document Library -> Library Permissions
2) Stop inheriting permissions
3) Select all permissions and choose 'Remove User Permissions' from the Ribbon
4) From the Ribbon, select 'Grant Permissions'
5) In the dialog, select your SharePoint Group and select the appropriate user permissions from the 'Grant users permissions directly' choices
Now try to login with less privilege users, you won't see list/library displayed in site content.Same is case with Quick Launch Also
The Quick Launch displays only the libraries that the logged in user has access to. Other libraries are automatically hidden.