we have a good relationship with the children we raised, along with their children. Our dog, however, is always with us and it just feels good to watch after her. We don't consider her a child, just a very good, non-verbal friend.
There is certainly an imbalance between dog and human authority/autonomy/agency, but that is not the only dynamic in the relationship. And it’s not necessarily the defining dynamic, nor is it consistently applicable.
Whether she understands how we feel about her isn't the point or the definition. She seems happy and doesn't live in any kind of fear, and that makes all the difference to us. Anything else is just picking fly scat out of pepper.
The parent-child relationship is asymmetric in ways that are often not as visible to the child as they are a parent. There's a reason why for generations, parents have been responding to their childrens' arguments with "when you're a parent, you'll understand".
My comment was another counter example to the GP's statement: "if you can be loving and empathetic to a dog, you can surely began to be that way to humans."
So the point is loving dogs doesn't necessarily correlate with being loving towards humans. That doesn't say anything about what loving dogs does correlate with.
That's very different from what you seem to be implying.