Folks, wouldn't you know, but minutes after posting my hopeless situation I have found an important clue. I should have asked my cat.
The Security Logins for the instance has a Login 'BUILTIN\Users'. If I add 'BUILTIN\Users' to Security Users for a particular database and give it the db_owner role, this fixes the problem for that database. However, this is ugly and I'm looking for an elegant global fix. I created a domain group called 'SQL Admins' and tried to add it as a Security Login, but the group doesn't appear in the selection list. I was hoping to use this custom group as a way of assigning specific users the permission to own and manage databases, but it doesn't work as I expected. There probably is a proper way of doing this, but I haven't found it yet. Greg