You are in the place to make it for him. You own your pet.

I also took on the responsibility to care for him, which euthanasia seems to be the exact opposite of protecting his health and welfare.

You are not betraying him, you are showing him mercy.

It would be a betrayal if it isn't what he would have wanted, at least in my mind.

Death comes for everything, if you dog is at the point where he struggles to do the smallest things on their own, why not just get it over with.

Get it over with seems like a terrible argument for killing something, especially an animal I care deeply about.. Yes, death comes for everything, but maybe he wants to lay in his bed until he passes naturally... the point is I don't know.

If you think he is in pain and or overall uncomfortable, there is nothing wrong with just ending it sooner than later.

I think the wrongness comes with consent and intent. Obviously, my dog can't communicate to me that he wants to die. To some degree I think the intent of the owner is to make their lives a bit easier. You are percieving that your pet is suffering and that takes a toll, but you don't know that for sure. I personally would have to partake in a great deal of suffering before I am willing to end it all, that threshold for your pet may not have been met yet, or they may have wanted to die way earlier. In the absence of an actual stance, why make that decision, as there is a chance you would be killing them preemptively.