Enable debug mode and maximum error reporting in Joomla global configuration then retry to see if any errors output. Only reason for that is an error, something interrupted the CB user store, or a redirect before the CB user store.

I was just testing CB Core Fields Ajax while working on CB Privacy and worked fine across the 6 different fieldtypes I tested. If you're seeing "Error Access Denied" then you've a permissions problem. Either the user doesn't have endpoint access to their profiles (make sure Profile View Access Level is set right in CB > Configuration > User Profile) or they can't access the menu item it's using to submit the ajax request (typically this is just the profile menu item).

You should have a profile edit menu item at all times. It should be set to Public or Registered at the very least. CB will check access to profile edit on its own. It does not need to be on a visible menu module though.