I suppose that argument could work, if the nature of the child's disease was explained. However, it wasn't. The scenes with the family were perfunctory handled, and I felt that they were a plot device.
Not only "could" that argument work, it does
The scenes work perfectly to establish the curative nature of Khan's blood. Any more detailed explanation is completely unnecessary.
Of course those scenes were "a plot device." What else were they supposed to be?