First, the victims generally will still be upset. They may have medical or veterinary bills that need to be paid, and they likely will still feel the trauma of the situation. They might feel that "justice has been done" when you destroy your dog, but it won't stop them from suing you anyway. I have seen far too many cases where a dog owner gets sued after the dog has been destroyed to believe that victims are placated so quickly.
Second, it does not affect your liability. In Arizona, a dog's owner is strictly liable whenever the dog bites someone, unless the person was trespassing or provoked the dog. There is no defense to liability by demonstrating that the dog is now dead.
Third, it will prevent you from gathering evidence that may help you. As noted above, provocation is a defense to liability for a dog bite. The Arizona legislature has decided that provocation should be decided based on what a human thinks is provocative, and not what a dog thinks. While this is a genuinely stupid law, it still allows for education of the judge or jury as to what a reasonable person should think is provocative to a dog. Here, an animal behaviorist can be helpful, by explaining what provokes a dog and why. The behaviorist also can evaluate your dog and offer an opinion about whether or not your dog was actually provoked during the bite. If your dog is dead, however, the behaviorist will be unable to do this.
Finally, and perhaps most importantly, it will make you feel like a terrible person. If your dog is dangerous, a judge will likely order that he be destroyed anyway. Don't volunteer for this painful task -- leave this decision to someone whose job it is to make tough choices every day. Then, when your dog breathes his last, both you and he will know that you tried to spare him. It may be small comfort in such a sad situation, but it beats the guilt that comes with knowing that you discarded your friend when he became a liability.