I can see this, also as he is promoted - I imagine a Lieutenant might still be able to be "one of the guys" in a way that the XO just can't be (for newer people) - but I also have trouble seeing how you wouldn't remain friends with the people you did know before, i.e. Uhura. Maybe just "You have to act professionally when other people are around", which also could be a discussion that happens just after she was flirting with him on the bridge (and would make that whole thing in TOS make more sense in line with the other romantic instances for Spock - they never were romantic actually, it just was the last time she was teasing him on the bridge, and she happened to do it that way that time).