My testing shows that it's the SID cookie that holds the login info, not the NID one. What's odd is that after adding that to my whitelist and relaunching Safari, it appears that I'm logged out of YouTube again. But, clicking on the "sign in" button logs me back in without having to enter name/password. It's as if the site doesn't automatically recognize the SID cookie. It has to be directed to it by clicking "sign in". Something like that - I don't know exactly how it works.
Thanks for the help!