The solution to dogs having aggressive behaviors is never to harm the dog. If your toddler bit someone, you wouldn't say that the toddler is a danger to the human race and kill it. Instead, you would teach it better ways to express itself than violence. Pitbulls deserve the same treatment.
The fundamental error with this line of reasoning is that humans are different from animals. A pitbull is a) significantly more dangerous than a toddler, and b) an animal whose life is less valuable than the toddler.
If people are allowed to own animals that are highly dangerous if trained accordingly, they should assume full responsibility if their animal goes apeshit. If the dog murders a three year old, the dog owner is held accountable. That's the responsibility of owning a killing machine.
Putting it down might be the only realistic option sometimes, but I disagree that it has "no righteous place being alive at all." Righteousness is inherently subjective. One could argue that killing an animal that is not at fault for what it is is an entirely immoral act. I might agree.
A vicious and maliciously trained pitbull is fundamentally a danger to society. If a pitbull is trained to be this way, rehabilitation must be shelved for euthanasia. The risk is simply too high to allow a potentially deadly animal to reintegrate into society.