You're right, the setting's not there - which is doubly weird because the code to use the setting absolutely exists in the snippet I mentioned.
Sure enough if I add the config, that works as you'd expect.
$standard_fields = array('website', 'posts', 'warning_status', 'date_registered');
$fields_no_registration = array('posts', 'warning_status', 'date_registered');
$standard_fields = array('website', 'posts', 'warning_status', 'date_registered', 'action');
$fields_no_registration = array('posts', 'warning_status', 'date_registered', 'action');
and
// What about standard fields they can tweak?
$standard_fields = array('website', 'posts', 'warning_status', 'date_registered');
// What fields can't you put on the registration page?
$context['fields_no_registration'] = array('posts', 'warning_status', 'date_registered');
// What about standard fields they can tweak?
$standard_fields = array('website', 'posts', 'warning_status', 'date_registered', 'action');
// What fields can't you put on the registration page?
$context['fields_no_registration'] = array('posts', 'warning_status', 'date_registered', 'action');
Plus a language string, say in ManageSettings.english.php:
$txt['action'] = 'Profile Action';
And presto, the thing turns up as a setting which is honoured by the profile page. I'd add a screenshot but apparently a 9KB PNG file has a problem 
@Spuds can put the above into a PR for you if you like? Seems like a legit bug fix to me to have a thing respect configuration but not provide the configuration...