We have had about 10 cats over the last few decades. Now we have four (three whites, one more or less black). And yes, over the years we lost three cats to traffic right in front of our house. After number three, our kids even organised a neighbourhood protest with all the other kids in the street and started to jump in front of cars and attack drivers with leaflets and angry language.
After number three I stopped getting cats in the house. I mean, how many dead cats can a family stand anyway as for kids its heart breaking. But many years later a friend, who owned a cat already, found a sickly stray one and took care of it. Unfortunately this stray was not accepted by the cat in the house. So I took it over. And we had years of fun with it because it appeared to be very clever and very fond of people :-).
We did start with a “hell and doom for cat's” programme what to expect outside the front door. It worked. All cats we ever had since the stray one know we do not want them to leave the house through the front door. This makes them nervous about what to expect and they do not frequent the front garden as much anymore. So, we have been free of cat accidents since.
We did loose two very dear ones to cancer also. So, all our cats left us either to accident or sickness. O, or as kittens to relatives :-))
In the end one never gets used to it, but the detachment needed to take the proper actions in the best interest of a terminally sick/wounded pet has grown upon us.
It makes your stomach turn to see the doctor push his lethal needle in your beloved pet, but it's my wife and me who are holding and caring for it (in tears and silently) during that moment. They are in trust, calm and full of dignity when they leave you, ending their suffering at once.
I must admit, the dog was the most hurtful as you said, I knew him for almost all my life when he died. In its own way, he was a family member with most of the trimming that comes with that :-))
Dogs go crazy with joy when you come home, hehehe. They can’t express their love for you enough :-))