For the benefit of others, I figured out the solution. And yes all of the iPhones involved satisfy the requirements to be a recovery contact. The problem was related to a similar problem we were having:
iOS insisted on associating recommended contact with their email addresses rather than their mobile phone numbers.
This contact/e-mail issue is the root cause for this particular problem.
This is the fix:
When selecting a Legacy or Recovery contact, do not select the recommended one, instead, select Choose Someone Else. Then on the To: line, use search to find your contact by entering their name. If the info below their name is their mobile number, select that contact. If the info below the contact is an email address or the incorrect number, select the ">" button to the right of the contact. That will present a list of phone numbers and email addresses. Select the contact's mobile phone number. Click Add and continue following the instructions to add that contact.
The same process can be applied to iMessage, which will happily attempt to conduct a chat with a contact's email address without informing you it is not using their mobile number. The fix is to start a new chat, search for the contact as above and follow the same procedure to select their mobile number.