Is it that the dog dies faster than it would have?
That's not actually an issue. We already have vets who just put terminally ill pets down.
So, who does the bureaucracy help? Who's protected by it? Why is it there at all?
Part of his filings will be actually stating the "terminally ill" part and having this approved by an ethics committee. Making a moral judgment here is the committee's actual role as not all cases are so "simple".
It could have been a single informal paper that says "the animal is terminally ill, my judgement call is that this is unlikely to cause excessive suffering and might help instead, even if the chances are low, and if my judgement is proven wrong and this appears to cause excessive suffering the animal will be put down humanely". Signed by the veterinarian and the owner.
Because the system is high speed low drag, and trusts the veterinarian and the owner to make reasonably good calls about pet health and suffering - unless proven otherwise by overwhelming evidence. The system trusts people by default, and that 3 months long process and an ethics board come into play when there's a suspicion that this trust may have been abused.
Of course, that's not the world we live in. Which is why we're having this conversation.