I am posting my reply to the thread below as a new idea, to highlight my take on the OP's original description, to garner clarity.
The widespread use of CardDAV has grown significantly, especially for businesses.
It's frustrating not being able to set a remote address book as the default, in Thunderbird's settings.
Whenever adding a new contact, particularly from a received message, when clicking on the address book icon, it will use the local Personal Address Book by default. One has to then Edit it either there, or drag the entry between the Personal and Remote address book within the Address Book manager.
The thorough solution would be: