I added a new local user on all the systems of the same credentials and then granted that user access to the shared areas. As expected, using those credentials granted access to the shares from all systems to each other system.
However, I should not have had to do this. These shares have been working fine for over four years relying upon upon the primary account credentials on each system to access the shares since: 1) each system has this same primary account, and 2) the shares on every system are accessible by the primary account (by default). This new user will never be used for local login only for remote login (accessing the shares). There is nothing special about an MS account over a local account in this regard; the use of the email address as the account name and granting MS ability to rely upon that account is the only difference. It should not affect access to network shares; at that level, it is just a user account.
It's always exciting when things that have been working one way for a long time decide to stop working that way with no clues. In my experience, Windows Shares has regularly exhibited individuality in this area.
But, I'll consider this closed for now.