^Again, I'm not making a black-and-white, all-or-nothing point here. I'm saying it's a matter of degree. We'd try to protect our pets, but wouldn't be likely to, say, risk our own lives to save a pet goldfish, or spend thousands of dollars on a kidney transplant for a sick bunny. There comes a point where most of us would put our pets to sleep and move on rather than go to the kind of extraordinary lengths we'd go to for a sick human.
By the same token, while there are certainly circumstances wherein Q would help Picard or humanity, there are probably limits on how far he'd go. He's seen countless species live out their lives and go extinct; to him, it'd be part of the natural order of things. So there are probably circumstances where he'd consider "putting us to sleep" to be the most humane option, where his focus would be on easing humanity's inevitable end rather than keeping us alive indefinitely.
And if screenwriters or fans want to anthropomorphize him and make him care about human life exactly as much as we do despite his profoundly alien perspective, I'd consider that to be illogical and out of character as well as a failure of imagination. A being that different should
have a different perspective, whether we want him to or not.