How to expose edit link on editable areas for all authorized users?
There are two ways to do this.

The "gears" feature is from the "Use contextual links" permission on admin/people/permissions.  If you give that to the appropriate roles, your users should see that.

Personally, I don't like the Contextual Links feature, mainly because the markup it adds can mess with theme layout when you have other positioning going on. I typically add in my own edit links to pages, usually below the content somewhere. I will sometimes use a display: none on them, and then have a link in the footer menu that shows them all via jQuery.