I did some checking on your site, and I exported your Access settings and imported them to my own test site to see if I could identify any problems with the custom role that prevented it from seeing edit post links, but it worked correctly there.
I was going to try testing the visibility of the edit links on your site by creating a user with a heffstaffadmin role, and then I noticed that there are only two users on the site, neither of which has that role.
You have a user named hefstaffadmin, but it has the role of author.
So, did you actually test with the role of heffstaffadmin?
I can see that the author role is also set to be able to edit the posts, so the edit links should appear, but I want to clarify what exactly I should be testing.