Maybe she just is aboard another ship because of a matter of different timing.
One of the key differences with tos is the fact that, in this timeline, Spock and Pike started working together on the Enterprise years later compared to tos. It seems it took them longer to launch the flagship under Pike's command, so maybe Number One accepted another job while Spock became the first officer and he was working at the Academy while the ship got completed (instead of taking a job on another ship).
It is worth to mention that according to the official app, he enrolled in Starfleet the same year Uhura and Kirk did but graduated sooner. So it seems he really was just a graduate student who got asked to teach some classes (eg the one Uhura followed). Tbh, I got this impression from the movie as well because 1) he doesn't seem to be much older than Uhura and Kirk (3 years of age difference between them is believable) 2) during Kirk's disciplinary hearing Spock is introduced as 'one of the most distinguished graduates' by the Academy President. The tone used, combined with the fact Spock may have a commanding presence but he obviously is as young as Kirk and Uhura are, seems to suggest that he has graduated recently himself (with the Kobayashi Maru test perhaps being a project he had started when he was still a student and teacher assistant).
It also seems tos Spock joined Starfleet when he was much younger than kelvin Spock. This may explain why his career here starts around the same time of the others. Pike also seems to be older here, again because his command on the flagship started later.
Generally speaking, I think Discovery is setting up even more differences between tos and the kelvin timeline.
For instance, does kelvin Spock also have a human step sister? It doesn't seem so; they also seem to ignore Sybok, somehow respecting, so to speak, Dc Fontana's original plan that Spock was the only child of Sarek (I remember the writers said they used her books for reference and we know that she hated stv its guts for giving him a bro lol )
Maybe Michael's parents didn't die and she had a healthy childhood with them instead of being adopted by a dysfunctional human/vulcan family. Or she got adopted when she was old enough to not get too influenced by them and she and Spock could be friends. Maybe kelvin Sarek wasn't THAT hypocritical with his son, and Amanda was a 'real mother' and the first one who made Spock understand empathy and his human side, after all.
Did Kelvin Spock too go 'insane' because of red angel visions? Again, doesn't seem so, and he already is past the age where he'd have those visions.
This Spock's only conflict is the usual human/vulcan one only with the added stress of survivor guilt that, along other differences, accelerated his emotional development.
In fact, what many fans seem to miss is that he doesn't NEED to become like tos Spock was because he has already resolved some issues it took tos Spock years to resolve. For him to become like tos Spock was at the beginning would mean going backwards, an involution of sorts.
He's Spock in spirit, but he also is foundamentally different and has his own integrity. Maybe for all his struggles, including the loss of his home planet and his mother (a terrible thing his tos counterpart didn't experience), this Spock's childhood and his own life experiences (last but not least the fact he fell in love and he is in a relationship too) ultimately made him a more balanced person.
Of course, because discovery is retconning his tos counterpart's backstory, it seems kelvin Spock is even more different now but we know that, originally, his story more or less followed his canon before discovery added stuff that wasn't there before (so the writers couldn't even think about putting those elements in their movie).
But I digress. I think that, however unintentional, Discovery is adding even more interesting differences between the realities which, again, isn't an issue for this trek because unlike Discovery, it doesn't have to reconcile with established canon that much beyond using it as inspiration. The fun part, here, is precisely trying to imagine what went differently and why.
