Seems you have swayed me just a little bit. This is how I'm looking at it, since Uzumaki at the time had rule of Kurama (as he was used as a tool as stated in the manga), while not being Kushina's birthright, she was currently in possession of something that had been a tradition with the Uzumaki since they had the capabilities to contain Kurama. She lost what she had previously owned as a tool and Minato passed that on to her (their son). I guess since we're looking at Kurama as a living being instead of a tool (at the time, what he was established to be) and the definition of birthright (a right that you have because you were born into a particular position, family, place) is why I am looking at it the way I am