There is a taxonomy with an image associated. It is supposed to display with the image and no title. However, in going to the "Manage Display" section for this taxonomy, the name field is not displayed at all.
Searching for this has proven futile. I can manually remove it by modifying display code but then that leaves no way to change its display from the CMS.
Why is the name not manageable as part of the display?
Because it's a 'property' of the entity, not a 'field' as such. The 'Manage display' tab is really a 'Manage field and pseudo-field display' tab, it doesn't take properties into account.
This is completely standard and is inline with all core entities. For example, if you check the display settings for one of your node types, you'll notice the 'Title' isn't available there either.
Is there some way to manage the taxonomy term display including the name?
The simplest way is to manage it with templates, as you've already suggested (and again, exactly how it would be done for all other core entity types).
You could always implement
hook_field_extra_fields() to expose the term name as a pseudo-field on the display forms; but bear in mind you'll have to link up the logic to hide the titles from the template file yourself, as the core taxonomy module doesn't have that functionality.
I think Add entity properties to "Manage Fields" form is a relevant D.o discussion, there's even some sample code to get you going.