I feel the same way. I like the unfolding relationship drama because it feels like a relationship to me. Its emotions and connections spilling out because some things need to be said.
I don't find it nagging, because I see human relationships unfolding. Kirk is very impulsive and emotional and loyal, all of these things to a fault, and it can come gushing out if he isn't being heard. With Spock, I think Kirk feels the need to prove himself, starting with the KM simulation. I think that interaction, how Spock goads Kirk, sets the tone for the beginning of their relationship, and moves on in to a mutual sort of respect, with a lot of teasing.
Seems to me like Kirk and Uhura feel like they need to say that Spock should be more human for them to like him.
So really how different are they from the bullies on Vulcan tormenting Spock because he isn't Vulcan enough?
I never get Uhura's problem with Spock. If she doesn't like him then she should get another boyfriend.
If Kirk wants a First Officer who lies on reports then he should get someone else. I don't think 'teasing' involves Kirk making racist remarks to Spock. Spock could just go to New Vulcan to live if he wanted to be 'teased' with racist remarks.
Uhura has concerns that her boyfriend would rather die and doesn't want to talk to her about it. That is a point of discussion in any relationship. I don't see it as unreasonable to not discuss the possibility of death on a Starfleet episode, but Spock is reporting a desire to die. Um, that is concerning regardless of racial tensions. It isn't that they want him to be more human, but they want to understand him. That isn't always easy, and can make relationships far more tenuous if there is someone who looks at things from a strictly logical point of view.
Kirk doesn't want Spock to lie, on the face of it. Kirk wants Spock to be loyal to him first, because that is how Kirk is loyal to Spock. Kirk is loyal to a fault, putting friendship before duty, while Spock is the opposite, putting his duty to Starfleet above all else. It's a conflict, and to me, is what makes the relationship interesting.
Also, how are the racial remarks, and you'll forgive if I don't remember all of them, different from McCoy's teasing in TOS?