I suppose that argument could work, if the nature of the child's disease was explained.
You wanted to know which disease she had specifically ? What difference would that make ?
None, as Kirk was brought BACK FROM THE DEAD, rendering the poorly kid plot point entirely redundant. You could lose that aspect and lose nothing as McCoy injects the tribble in an entirely unrelated
If it had been brought up as a reason as to why
McCoy decided to inject the tribble with Khan's blood it would have been pretty clever but as it stands you have one scene which sets up nothing and another which comes out of nowhere.
It does not logically follow that if Khan can save a poorly child he can resurrect the dead.
That scene was for the audience. The power of his blood is telling us right up front that there's something different about Harrison. The scene also shows just what kind of cold, hard, manipulative bastard Harrison is. Even the girl who was saved will be affected by Harrison. She was a pawn, too. Think she'll have a care-free life going into her old age carrying the memory of what her father did to save her?
As far as it not being logical that Harrion's blood can save Kirk just because it saved the girl, this has been hashed over a lot on these boards, but a world of transporters, warp speed, and all kinds of other pseudo-science is far less believable when you think about it than invoking a theory from cryonics that says as long as the brain has not completely lost its ability to function, a "dead" person can be resuscitated. McCoy told Kirk he wasn't "that" dead. Maybe that was only half a joke.