Makes perfect sense. If I transfer a bit of software from my computer to yours, it is likely to remain fully functional in both. It would take very specific effort to remove it from my computer.
That begs the question though: Does that mean that a Vulcan can simply "upload" a copy of themself into any host, willing or unwilling, or even into multiple hosts and have then several copies of themselves running around? Is the katra not unique to the individual and a "one person at a time" kinda thing?
Hardly all. At most everything until the point of upload. And probably just the "essence", since this word was used by the Vulcans themselves to describe the katra business.
Well, McCoy seems to know how to operate the science station equipment on the Enterprise, something I would assume he wouldn't have known before. And if that's the case, then it would seem like McCoy got enough of Spock's knowledge and memories to have detailed technical information. That seems, to me at least, a bit more than what I'd describe as "Spock's essence."
And we have no evidence that the regenerated Spock would have understood a word of any language until upgraded with the katra of a compatible adult (that is, the one from Spock's earlier, older self). Soothing sounds, yeah. Touch telepathy messages, perhaps. But not words.
I always took away the impression from his conversation with Saavik that the regenerated Spock understood what she was saying, particularly when she asks if he trusts her. I could be wrong, of course, but that's the impression I got.
That makes the best sense of all. If Spock downloaded his life's story to McCoy in the engine room of the about-to-explode Enterprise, then when McCoy dumps this back into Spock's noggin, that's where Spock's memories are likely to end: he'd remember next to nothing from between his death and his awakening after the mindswap.
It seems somewhat unclear how much or how little the "rejoined" Spock got back from the katra. We see in the beginning that he has to struggle to even remember the names of his closest colleagues. And we see him having to be completely re-educated and retrained, despite the fact that when his katra was in McCoy, it still seemed to possess its technical skills.
But then by the time of TFF it seems that Spock is back to his old self, complete with memories of his childhood, his long-lost brother, and so forth. Did he relearn all of that, or did his own memories from his katra just take a while to reintegrate themselves? And, if the latter, why all the rushed retaining? Why not just let him take the time to get his own knowledge and memories back?