Yeah, I actually found that part of the plot effective and easy to relate with coming from my own experiences of loss. John is basically transferring all the pain and grief from losing his wife onto the puppy, and it gives him an outlet for his grief in the form of revenge. People criticize Keanu Reeves’ acting ability but the “I’m thinking I’m back” scene works for me.
Also, John Wick is such a badass of a character that they sort of need to overdo the part that makes him sympathetic to the audience or else he would just be terrifying. The movie actually plays on a lot of villain tropes for John Wick, like in all the scenes where the gangsters are terrified of him and telling stories about him. If you take out the part about the wife and dog, John Wick is basically a slasher movie where John is the vicious serial killer.