In general, ‘upload_files’ is enough to see all items of the ‘Media Library’ and may add a new one.
I deactivated URE at your site temporally and ‘subscriber’ role still did not see Media Library items. This showed that some other plugin or even theme blocks access to the media library items. After some testing I discovered that your site current setup requires at least the ‘manage_options’ capability to be included into the ‘subscriber’ role in order it has access to media library items uploaded by others.
So I added ‘manage_options’ to the ‘subscriber’ role and blocked all extra menu items with ‘Admin menu’ add-on. Please check.
