Spock isn't Spock. He died from radiation exposure after defeating Khan. He just implanted a copy of all his memories (i.e. "katra") into McCoy before he died, in the hopes that McCoy could get them transported into a katric ark and put on the shelf. Vulcans have a nifty system where they can access katras of deceased individuals, probably for a set length of time. Must be a boon for historians.
Surak's katra existed in living individuals (Syrran, Jonathan Archer, some other guy), instead of an ark, perhaps to extend its "lifespan" or just due to its importance.
Sarek seemed to think that Spock's katra may have lingered, somehow, in his body after death. Maybe there is a way of accessing these telepathic brainwaves shortly after death, especially if the body is frozen or could be partially regenerated enough to recover some memories.
Spock's katra was inserted into a fresh body, the body of the weird clone regenerated by Genesis protomatter, displacing or more likely merging with the Genesis Spock's consciousness. Legal fiction of the time treated this new Spock, who had all the memories of the old Spock, as the one true Spock, and he received his rank and privileges from Starfleet probably after the Whale Probe Incident.
This new Spock, of course, wasn't much of a Starfleet officer, and, like his father, quickly pivoted to diplomacy and ambassadorships, with a keen focus (beginning at Khitomer) on reunification with the Romulan people. A hugely lofty goal, and perhaps psychologically linked with the odd reunification of minds that brought new Spock into existence in the first place.
Spock's was notably a very rare case, something that hadn't happened in centuries. Surak's katra and the katric arks dating back to pre-Surakian times shows that this is a telepathic technology with a very long history, and cases probably exist of warlords or something living for multiple centuries by implanting their katras into young sacrifices and fully merging their personalities.
By Sarek's time, katric arks were the preferred method of storage, but I have a feeling that even they were neglected in the couple centuries leading up to T'Pol's time, when telepathy and mind melds were shunned in favor of cold, reasonable logic.
Sarek grew up in the post-Reformation age, when the Old Ways of Surak were brought back and studied. He kept a large vault of katric arks, and practiced melding regularly. It might not be societally universal, yet, but it is a practice of his family to store and maintain these arks. Spock might have been Sarek's first chance at making an addition to these arks, if Skon was still alive and Solkar died at a time preceding the Reformation.
100 years later, the fascination with katric arks and maintaining a katra may have ebbed away again, and Sarek's private thoughts are properly kept private while his public actions are recorded in the Federation databanks.
It had only been a couple of generations (in terms of Vulcan lifespan) since the Katric arts and sciences had been rediscovered after having been suppressed for so long. So even at this point, there may still have been some wrinkles to iron out in the whole process. An incorrectly performed Katric transfer might possibly be a traumatic experience for all parties involved.
