Admin user can't manage fields on custom content type
I feel so stupid. But I'm leaving this answer so that the next person to have this same problem will save 3 days of development time:

The module that allows fields to be created on entities (nodes, user etc.) is creatively called "Field".

But, that doesn't mean you'll have a UI to create/edit them. There is another creatively named module called "Field UI". If this modules isn't installed AND enabled you won't have a user interface to edit your fields...