Same here.
But -- as you added, the joy of watching our children grow up into successful and decent adult humans made it all worth it.
Now that all four children are out of the nest, I finally have time to indulge my dog. So yes ... he's my baby. And what a joy he is, my little rescued mutt.
Although I have no intention of owning a dog, I think that your arrangement is great; as you’ve already had children, you will NEVER think of your dog as a person (even if you love it very much). I don’t mind when people have pets instead of children; that is their right and decision. I have the right to be disgusted and amused when they treat them as humans (or better), and ascribe human attributes to them that aren’t real.