This really sounds like the same problem I had with version 1.0.x. About an half year later I finally found out what I was doing wrong. It appeared it was not a bug, but my misunderstanding of how things work in Joomla.
What you have to do first is create a new menu item of one of these specific types:
Blog - Content Category
Blog - Content Section
Blog - Content Category Archive
Blog - Content Section Archive
Let's go with "Blog - Content Category". Give it a name, select a specific category, and set its parent item (use Top as parent just for testing, you can move it later). If in your global config "read more" is enabled you don't have to enable it perse overhere, to be sure you could set it to "show". Save it.
Now go to 'Content' -> 'Content by Section' -> 'the section which is linked to the category choosen above' -> 'section items'. Overhere create a new item, give it name, select the right category, put some text in the intro input box, put some text in the read more input box and save it.
Now open the website, click on the menu item you created above. Now the read more button should show up.
What you probably did before (like I did ), create a content item and linked this directly in a menu.