"It caused me to half-remember a story. I think it might be by Eric Frank Russell, and no, it is not Allamagoosa.
In this story, the pet dog of a member of an exploration crew gets abandoned when they have to do a panic take off.
The dog is found by an alien race who study humans through it’s knowledge of humans. They discover they like dogs and because of the dog’s feeling for humans, they are kindly disposed to humans. There is an eventual first contact and the dog goes home. “How could we be hostile to someone who returns a beloved pet to it’s Master?” A really good story and crowning moment of heart-warming."
Definitely by Eric Frank Russell. It’s titled, “Into Your Tent I’ll Creep.” I have it in an anthology called, “The Best of Eric Frank Russell.” Allamagoosa is in there.
Oops! Sorry, mis-read your description. The story you’re looking for is by Eric Frank Russell, I just can’t recall the exact title and don’t have the volume handy at the moment. The story I referenced has a different outcome.
It’s not a Russel story. I remember reading the story the OP is describing and Russel’s Allamagoosa around the same time, in a sci-fi anthology that was not specific to Russel. Upon further research, I found Great Stories of Space Travel from 1970, which includes Allamagoosa and Propagandist by Murray Leinster, which is the story of “a dog named Buck” who gets left behind by a human space crew, to be discovered (and mind-read) by aliens, who ultimately conclude that since Buck is okay, and Buck likes humans, humans must be okay.
I would say that the OP is thinking of this, very good story. It was in an anthology that I got as a gift from an aunt or second cousin, or someone, that first introduced me to science fiction a long time ago.