That is a dangerous line they are walking if they do it without owner's permission unless they have it outlined in the contract that they will do one if anything should happen to the horse. The horse may be in their care but they are not the owners. Many owners would be fine wit it - but it only takes one to decide to sue over it.
With humans, if there is anything suspicious about a death, the medical examiner mandates an autopsy. People would be able to hide all manner of murder if not. Trainers too.....