Dogs also have no mindset for emotional surrender or giving up. They have no awareness of the inevitability of death as we do and they have no fear of it. It is fear that so often influences and aggravates our perceptions when we are sick or dying and it becomes impossible to separate the fear out from the actual illness after a while. But that's not the case with dogs. Whatever we observe to be wrong with our sick dogs, it's all illness. And we don't even see the full impact of that until it's at a very advanced point, because it's a dog's nature to endure and to sustain the norm at all costs.
If that includes pain, then that's the way it is. Unlike us, they have never learned that letting pain show, or reporting on it, may generate relief or aid. So they endure, assuming in their deepest doggy subconscious that whatever we abide for them is what is to be abided. If there is a "look in the eye", or an indication of giving up, that we think we see from our beloved dogs, it isn't a conscious attitude on their part or a decision to communicate something to us. It's just an indication of how tired and depleted they are. But they don't know there's any option other than struggling on, so that's what they do.
We must assume that the discomfort we see is much less than the discomfort they really feel. And we do know of other options and it is entirely our obligation to always offer them the best option for that moment, be it further intervention, or none, or the gift of rest.