Menu link that uses title from router item's title callback?
Menu links that are registered through hook_menu() do this already, so if you are using the system-provided menu link, it should pick up whatever you have altered "title callback" to be instead.

The system-provided 'user' menu link is being altered by user.module to prevent it from showing up to anonymous users (see user_menu_link_alter(), user_translated_menu_link_alter()). That might be getting in the way of what you want, in which case you could bypass/unset the user module implementation in a hook_module_implements_alter() function.

If you're using a custom menu link created through the UI, I don't believe you will be able to make that link title dynamic short of implementing hook_menu_link_alter() and hook_translated_menu_link_alter(). That's probably more work than intercepting it at the theme layer, though.