I've had to have two of mine put down, one had cancer, the other had a thing called GOLP (geriatric onset laryngial paralysis) which is a nervous condition that started in her larynx, and spread down her spine till she lost control of her back legs. Both were around 11 years old.
You're right Andrew, having to do it is a horrible experience, and one you would not want to repeat.
Fortunately the rest of mine have gone in their own time, and it is easier, but only just. You still miss them, but at least you don't feel like a murderer (as irrational as that is), even though you know that you were only trying to save them from suffering.
For me, the only way to get over the death of one, is to get another. They never replace the one you've lost, but a new puppy requires so much of your time, that you don't have time to dwell on things, and I found that helps me a lot.
But I've spoken to others that have felt as you do, and that couldn't bear to go through the grief again. All I can say, is that if you get another dog, then yes you will have to grieve again at some point, but in the meantime you'll have years of company and enjoyment from them, and to me that is worth the price.