One often has to show a family relationship or other legitimate grounds to access a death certificate even though it is a public record and I suspect that this is the case in Nebraska as well. But anyone who has access to a death certificate has access to all of it, including the cause of death.
If the cause of death is initially undetermined, the coroner's office would have discretion about whether to update the death certificate as new information becomes available to the coroner, and a family member could make a personal appeal to the coroner not to exercise his discretion in this manner.
But there is no legal grounds upon which a family would have a right to go to court to prevent a coroner from updating a death certificate on any grounds. Keep in mind also that cause of death on a death certificate isn't even a complete sentence worth of explanation and is in very medical/scientific terminology.
Some government investigation reports are available for the public to see in an open records request under state law, but there is usually an exception for ongoing criminal investigations which is calculated to provide tactical benefit to the government in its efforts to catch criminals, rather than to preserve the feelings of the next of kin.
There might be grounds for the next of kin to ask a court to redact a government investigation report which would otherwise be available to the public in an open records request, and to seal the unredacted copy, either because leaving it open could facilitate identity theft, or because the material revealed would appeal to the prurient interests of third-parties reading it without advancing a valid public interest (i.e. if it would be basically pornographic for many people requesting it).