I DO think TOS Uhura and Spock are friends, but they're not, like, besties. Spock is seen ordering her around freely, as his rank allows him to do, and he speaks to her a subordinate. I don't think anyone except Kirk and Bones are allowed to be quite so chummy with him. It would make sense if Uhura kept her distance, knowing that Spock isn't much for emotional vulnerability.
The lyre scene, though...she's straight up trolling him, and the fact that she feels comfortable doing that to her superior shows the two must know each other a little. She knows Spock is not as menacing as he seems. She knows he's gay and she knows he'll find it funny if she sings about him being a womanizer. This is Uhura's way of saying “I know who you really are” because she knows him, even if Spock isn't willing to open up to her. But she is too polite to start bullying him like Bones, and she doesn't understand or love him enough to get him to slowly open up like Jim.
Idk, maybe it's stupid to find things like that in a TV show from the sixties (because that was obviously the last thing on the writers' minds) but I think Uhura has this stupid situation where you really like a coworker, but they've got too much shit going on to notice you, and I relate to that so hard. So knowing that SNW made Spuhura canon, it's...nice. Even for an eternal Spirk truther like me.