Our vet is very good in that he and a nurse will come to the house if need be, and the dog or cat can be pts in its own environment, which is often better than the trauma of going to the vet. He also has a side door, which enables you to leave with your deceased pet without having to walk through the waiting room.....
I was waiting one day, and a man came out of the surgery with a cat carrier. He looked very upset. Went to pay the receptionist, who can see the screen in front of her, which the vet fills in while you are there, took his card to pay, and then asked "Will you be bringing the cat back for further treatment?" The poor man looked even more upset as he replied in a choked voice that no, it was dead.
The strange thing is, I have sat holding plenty of my goats and horses over the years as they are pts, but, my husband has to take the dogs and cats, while I stay at home, howling!!
I am sure its hard for the vet and his staff, because its very likely they have treated the pet over the years, and dealing with a distraught owner is not easy.