Profiles using Participants (Event type), Participants (Event Name) and Participants (Role) custom fields are not available for online registration

Profiles that use fields from custom fieldsets that have "Used for" set to Participants (Event type), Participants (Event Name) or Participants (Role) are not available for inclusion in event online registration.

To reproduce problem:
1. Create a new event or keep in mind settings for an existing event
2, Add set of custom fields
3. In the "Used for" field select any one of the following:
a. Participants (Event name) and set to existing event from step 1
b. Participants (Event type) and set to event type from step 1
c. Participants (Role) and set to participant role from step 1
4. Add a field to the custom set of fields.
5. Create a new profile
6. Add the the field created in step 3 to the profile.
7. Open the event you created or kept in mind from step 1.
8. Click on the "Online Registration" tab.
9. Attempt to set a profile by going to one of the "Include profile" dropdown menus. The profile you created in step 5 should be available, but it is not.
10. Attempt a workaround. Click on "Create" to the right of the "include profile" dropdown. Create a new profile there. But then try to choose the field you created in step. It will be missing. Workaround fails.

These steps were confirmed on demo.civicrm.org. This was tested in CiviCRM 4.4.7 and the bug did NOT exist in 4.4.7.