'I realise that post mortems often have to be done, but by the time that happens, your body hasn't been breathing for a significant amount of time.'
Not true at all. If you fall ill and are taken to hospital and die quickly, there will often be postmortem to determine cause of death.
My niece fell ill suddenly at home, age 39. It was evening, the paramedics came in minutes and she went into cardiac arrest whilst they were there.
She did not recover, but had to have post mortem due to their not knowing what caused her to collapse (cause was determined to be myocardial infarction/hear attack/coronary artery disease).
There are several of us even here, in the bereavement section, whose children suddenly fell ill and the child died in hospital. Post mortem was required to determine exact cause of death.
My child did not require post mortem, but her cause of death was very obvious, she had been in ICU for nearly a fortnight before her death suffering from pneumonia and had been through stem cell transplant for treatment of leukaemia.
Still, after she died, we were given a form to register her death and another, and on reading them, I realised that if there is even a lick of conjecture, a post mortem is required.