Ok the component is still functioning properly, but the custom link class name format has changed so I had to make some changes to the field documentation in the component and the OP. Please update the theme component.
Unlike the custom menu sections, for the custom links, the names in the component settings should be the same case format as their actual custom names. Of course the default ones should still be lowercase just to make things even more confusing lol. 
For example, if I want to show a specific custom link called “GitHub Repo” for logged in users only:
I would enter the same link name in the show these links for logged-in only setting in the theme component:
So if you have custom links in the links display settings - show these links for staff only and show these links for logged-in only - you will need to change those component settings entries to match the link names.
Note that for custom menu sections (not links), the naming convention is still the same as it was before - it is still lower case slug style (ie: “Custom Menu Name” = custom-menu-name).
Thanks again for bringing this to my attention @nathank 
At some point I want to add the ability rearrange all the navigation sidebar menu sections, including custom ones, not just the default ones as it does currently. I know how to do it in hardcode, I just have to build it in to use settings and variables.