Coroners and forensic pathologists are two different groups of people. Coroner's deputies are death investigators (often part of a law enforcement agency, like a sheriff's office) while forensic pathologists are the doctors who do the autopsies. Both can "cut corners," yes - but in different ways. A Coroner's deputy might cut corners by not visiting the scene; by not examining the scene thoroughly, either in order to save time or because they are tired (many death investigations are at ungodly hours); by trusting the reports of the people at the scene about what happened without confirming whether those reports are accurate. The death investigation doesn't end when the deputy returns to the office and writes up the case. Frequently they have to complete their investigation, or ask others to, by getting medical records, police reports or questioning other witnesses who were not at the scene when they picked up the body.
A forensic pathologist might "cut corners" by doing an incomplete or partial autopsy; or by rushing through the case and by not following up with police or Coroner's investigators when the story being given does not match up with the injury on the body or the presumed cause of death. This takes time. Coroner's deputies get paid per shift and FP's in a coroner system get paid per case. There is no financial incentive for the doctor or the deputy to invest extra time in investigating a case. Many offices are understaffed due to budget shortfalls, so there is always plenty of work and not enough people to do it. A deputy has to be efficient with his/her time; so if a case looks like something routine, such as an overdose or a suicide, you might see them cut corners in the interest of working speed.
Yet speculation frequently follows an overdose. Generally, people who use drugs and alcohol are on the margins of society, hang out with unsavory or unreliable "friends," some of whom may have criminal pasts. This fuels funeral-parlor rumors that foul play was involved when the person dies of an overdose. Generally, the only way the police or coroner can confirm that foul play was not involved is with a thorough death investigation. But the investigative and autopsy findings also have to be articulated to the family. Coroner's staff are not necessarily medically trained and may not be effective communicators, so families might seek out other sources for answers. That other source is sometimes me - and I am always glad to serve as a consultant, but only after asking the person calling whether they aired their concerns with the Coroner's office or with the police. I also ask whether the family member had spoken to the original pathologist who did the autopsy. Frequently they have not. Sometimes if they go back and speak to those people who were directly involved with the death investigation, they will find out that the investigation was actually more thorough than they initially thought, or was not completely documented in the limited materials that were initially released to them.
Frequently I hear about families finding out about other peoples' suspicions at the time of the funeral. Unless the people fueling the speculation have direct knowledge of what happened, I would be cautious about putting too much faith in rumors. Family and friends often invent or exaggerate the importance of certain events in order to make themselves or others feel better about a death. I have investigated cases that were clear-cut suicides, where the deceased even left a suicide note, the family was understanding and seemed at peace, but then speculation at the funeral made them doubt the coroner's findings and suspect murder by an estranged lover or the roommate who found the body. Denial in the face of a death is a powerful (and expected) reaction - but entrenching that denial by piling doubt upon doubt is harmful to the grieving and healing process. I spend many hours counseling these families and I am grateful for my role in helping them find closure, and while I understand why well-meaning people can sometimes unintentionally cause more grief, it still pains me to watch.
What can be done about this? Well, if you are attending a funeral and have no idea why the person died, don't ask the family "what happened?" You'll find out soon enough. I love the Jewish tradition at a shiva (the equivalent of a wake) to bring food and not speak unless you are spoken to. When my father died the sustenance was appreciated and I was glad not to have to talk to anyone. If you feel really sad and want to share that with the bereaved you can always go with "I'm sorry for your loss," though I personally prefer sharing a happy story about something the deceased did that meant a lot to you or made you smile. That will have resonance and truly give comfort. If you are a family member and have been told things that disturb you about the circumstances surrounding the death of a loved one and are starting to have concerns, start by calling the officer who gave you their card at the death scene or informed you of the death. Tell them what others are saying, and ask them to help you have some closure. They may refer you to the forensic pathologist who did the autopsy, or maybe they will just reassure you that a thorough investigation was already done. If you don't get the answers you need, you can always ask to speak to their supervisor or to the pathologist yourself. In most cases, you will eventually be able to find someone to answer your questions, but please understand that death investigations take time. Just because you haven't heard back from the Coroner's office doesn't mean they have forgotten about you or about the case. The doctor may be waiting for the lab results before proceeding further. The Deputy Coroner may have received no replies when calling people to follow up on your concerns. Find out who the lead investigator is on the case, and what their hours are. If you call the office once every two to three weeks during their shift and always ask to speak to the same person, you will be able to get a progress report on what is going on with the case. I hope this helps you find closure.