When someone is going to die, there is no good way, no good time. But there are worse ways and times. As technology gives us more power to extend life of body, we are forced to considered what that means, and how to use that power wisely.
Three times I have had to decide when somebody's life ended. I think their are practical differences between deciding to withdraw care and choosing more active measures. But I think there is no moral difference. Either way, somebody vulnerable has placed their life in your hands. Either way, your only real choice is when to give up hope, and how much suffering they endure between that point and death.
I honestly don't give a fuck whether you're a better person for it. Or whether I am. It's not about us. The important question is whether you have done the best you can for the person. And "best" here, I think means something like "in accordance with their informed preferences, or as best as we can guess".
To that end, I strongly encourage everybody to get their families together and have real discussions about this. The Five Wishes framework is one good option. [1] The best thing my mom ever did for me was to have a frank conversation with me and my brother about her end-of-life preferences well before she was sick. Don't make your loved ones guess. And don't put yourself in the position of having to guess for them.
Personally, if I end up with something gruesome and terminal, I aim to die well before things get awful. Like Brittany Maynard, [2] I'd much rather go out on my terms than suffer pointlessly until my body fails, consuming absurd amounts of medical effort. Others can do differently, of course, but there's no way I'd want a doctor who frets they're going to die "a little on the inside" by following my advance care directive.
[1] https://fivewishes.org/five-wishes/advance-care-planning
[2] https://www.kqed.org/perspectives/201410280735/its-brittany-...
Are you saying this in a metaphysical sense?
Secondly, if you don't want to put your dogs to sleep, no one is forcing you to do so, just refuse it and allow them to live in pain till they die of old age. It's not the choice I would make, but then I don't have your views on euthanasia.
It's actually harder in some ways with pets because they can't talk and tell you detailed preferences for how they want to go when it's their time. All you have is the general guideline that, if their condition can't be treated and they are going to have to die, you want to minimize their suffering. I've had to do it with four cats, and we have two dogs now whose time will come eventually. It's not easy, but it's not like you have a choice; you made the choice when you took them into your life in the first place.
And don't forget that making the choice to let them into your life adds a lot more to your life than whatever the pain of having to put them to sleep takes away. Our two dogs are both rescues; we have given them many years of happy life (and many more to come) that they would otherwise not have had at all. And they have given us a lot of happiness in return. That will always be there whatever comes.