Thanks for the reply, Mary. I tried to add the permission on an assignment basis rather than site-wide and the results were the same. I was also using the Clean theme while I was taking the screenshots in my original post, so I don't think it's a theme issue.
I followed your steps regarding testing this on the sandbox and everything worked as it should: I was able to see the "Edit submission" menu item both as an admin and as a teacher. I then went to my test site and did the exact same thing: created a whole new assignment, set permissions, tested as both admin and teacher, and lo and behold, I can see "Edit submission". The assignment I was testing on was one that was older, created around the same time I made my test course years ago.
Can you think of a reason why the permissions would not apply to older assignments and only the newly-created ones? According to the Assignment upgrade helper, none of my assignments are legacy types.