That's because that clip is from the unaired pilot (later reworked into The Menagerie), in which Majel Barett's 'Number One' is supposed to be the logical, unfeeling character, not Spock. I think the viewer is just supposed to pretend that Spock isn't smiling in that scene.
We have a smiling Spock in Where No Man...
and was never an unfeeling character. And since pilot footage from The Cage
is included in an episode of the regular series run, you simply can't ignore it because it's inconvenient to your argument.