No matter who carried the katra, the only part of Spock that was left behind on Genesis was his deceased body.
Only if one assumes that somebody did carry the katra.
But in my interpretation (not necessarily the writers', but hey, whatever works), Sarek would barge in with the idea that Spock's katra ("him") had been left behind, since Kirk had not come forth with that katra. So logically Kirk either maliciously wanted to keep the katra, or then maliciously left the katra behind. Sarek would first believe the latter, but then decide to check on the former possibility as well, through the meld. Only then would the third possibility dawn on him: that Spock had been unable to transfer his katra. A typical emotional response from our second-favorite Vulcan...
Sarek doesn't speak of the body explicitly
. If we assume he doesn't want the body at first, then it's obvious why Saavik is not worried about the body, either. And Saavik wouldn't worry about the katra, either, since she'd be aware of the circumstances and convinced that Spock was gone, body and soul alike.
The katra is not canon therefore Spock and Kirk do not believe in it.
Vulcan mind melds are factual enough: they provide the melder with information he or she could not have obtained unless telepathy was for real. The Bajoran touch telepathy is a bit iffier, because we've never really seen unambiguous information transfer from that ear-gripping trick. Doesn't mean it couldn't be for real, too (at least for some skilled ear-grippers) - but the Vulcan version certainly is.
If melds are factual, then why not katras? Archer got information on/from Surak that he'd probably
not have had unless his telepathic communion with said deceased were factual.
Of course, Spock may still disbelieve in this katra stuff. It would just be a bit illogical for him to do so...