I figured out the solution to my issue.
Account A: Coordination Account (this was the account I used for myself and my wife--for years--for shared iTunes purchases).
Account B: my account (iTunes purchases pointed to account A)
Account C: my wife's account (iTunes purchases pointed to account A)
Account D: family account 1 (iTunes purchases pointed to account D)
Account E: family account 2 (iTunes purchases pointed to account E)
Account F: family account 3 (iTunes purchases pointed to account F)
Account A held the Apple Music Shared Family Plan.
Account A, D, E, and F worked fine. Account B and C were not recognized.
Solution: Changing the iTunes Billing Account (under Family Plan management for accounts B and C) away from Account D and to B and C respectively, solved the issue. It should not have to be this way.