"Is nuKirk closer to his crew than the original Kirk?" I'd say no. TOS was made in a different time, which I think plays a factor in how they portrayed relationships on screen. As to what was portrayed, I always looked at the fact that Kirk's crew chose to stay with him and keep a post that, as argued, may have held back their careers because of how they felt about their Captain. They respected him so much, that sometimes their choice was to follow Kirk and not Starfleet, which also affected their careers. TUDC portrayed that even once they did move on, they had massive amounts of respect for Kirk, ala Sulu's comment at the end with his whole bridge crew at attention 'Nice to see you in action, one more time Captain Kirk.' I was always left with the feeling that he was hugely respected by his crew, but that they didn't portray the more personal life side of that connection on screen.
They do show that more with the Kelvin verse, how they met, how they formed those connections and the friendships. Given a choice, I prefer the Kelvin timeline presentation, but I also love TOS as well.