To continue on with the dog analogy, we don't treat dogs because they can't verbally communicate with us (among other reasons). We can't use the same techniques to gauge progress with mental disorders with dogs as we can for humans. For dogs, it's flat out impractical. Further, we (as a society in Canada, according to our laws) value human life higher than we do animals.
In terms of this specific case, I haven't really read anyone on here posting with any real knowledge of schizophrenia. In Canada at least, doctors are extremely hesitant to diagnose someone with that particular mental disorder, and even when the doctor is leaning that way, it takes around 10 years of observation before they will diagnose. Treatment doesn't really start until that diagnosis. Meanwhile, the person dealing with voices in their head has to continue to live and work in society. With the voices in their head all of the time.
I have had to think about a lot of these things, as someone close to me probably is schizophrenic. They haven't been diagnosed yet, so we don't know for sure. Until they are diagnosed (which is probably never), should they be locked away? They are probably pretty high risk of doing something off the wall and harmful to others.
I agree with how Canada's laws currently work, specifically how they pertain to the case of the Greyhound Killer. The changes that need to be made is that MD's need to become more educated in terms of this mental disorder, and they need to be able to diagnose people with schizophrenia much faster, so that these people can get access to the limited treatments that are available.