They do want it investigated thoroughly.
Or rather, they are satisfied that the case has already been thoroughly investigated.
They simply don't want all of the (personal / private) material (in part, inadvertently) gathered during the investigation to be made public.
Like how would you like, e.g., video recordings of a coroner using a probe to examine a wound on your dead daughter being uploaded to YouTube?
Regards,
But what happens if this is actually a murder investigation and the killer(s) is caught?
How would the family, at that point, keep any information from getting out into the public?
If it's murder and they are claiming privacy, then they are keeping a killer on the loose.
If it's murder and the killer is caught, then privacy goes out the window as the evidence is presented in court.
All they would be doing is buying time.
-PJ