No, put the dog to sleep while it was in the vet hospital.
The dog was in a vet hospital. The vet will have said 'You really ought to put this dog to sleep'. The people who brought the dog in will have said (based on the video) 'No, we'll take him home and let him die naturally'.
So, the dog suffered for longer, unnecessarily, for what seems to be the benefit of warming some people's hearts thinking they were fulfilling some wish of a dog, and creating a soppy video out of it. It was exploitative and cruel - undoing all of the good they did by bringing the dog in the first place.
I think you have misunderstood what I meant, i completely agree with you.