Why can only administrators see blocks?
FACEPALM

ok, I found the ideal solution. When I looked at my authored nodes in /admin/content, it listed them all as Status: Published. HOWEVER, when I selected all of them, selected "Publish Content" from the action dropdown, and hit "Apply to selected items", suddenly everything appeared in the view block field.

My guess is that something in the database needed to be updated. Seems like a bit of a bug, though - this should have been automatically handled by either Drupal or the block_field module.

Anyway, fixed now.