I am entering an issue.
I just created a user with administrator level. This user can access the core teaser and main images and edit them. But the same user can't use the created field with Media JCE. They don't see the buttons to upload.
Ok, scrathc that. Had another look at this, and it looks like the Permissions for the specific "Download" field were incorrect. In the Permissions tab for that field, for the Administrator group, the Edit Custom Field Value option must be set to Allowed.
I have set this value on the site.
Just because you're not paranoid doesn't mean everybody isn't out to get you.
You were almost right. with your investigation, I found that the problem was on a higher level: in the com content permission settings, I had to change the "edit custom field value" to "Allowed" and all the fields are now allowed for that category of user.