You can to think about that. So it doesn't prevent gun suicides. The fact aside that someone can commit suicide with something else, the person doing it would be an authorized user of the gun. So no help there.
It doesn't prevent gun homicides. Again, these are done by authorized users of the gun, or people who have time to modify the gun. Remember for all the clever electronics, in the end guns are mechanical devices. So ultimately the electronics have to be something that mechanically disables the gun like a standard mechanical safety. A trigger disconnect, a firing pin block, that kind of thing. Ya well those are dead simple to bypass. So no help for stolen guns, the criminals would just remove the safety.
It doesn't prevent accidental shooting by any authorized user of the gun. Since they are authorized, it will fire. So any drunken games, etc, are still just as dangerous as they were before.
Already here we have, by far, most of the shootings that happen.
It may not prevent shooting where a gun is taken away from someone. Depends on how it works. If it has some way of reading the fingerprint when the trigger is depressed, then ok it could work. However if it works like a safety where you disengage it when you grab the gun, it'll still be disengaged if someone takes it away.
It would prevent accidental shootings where an unauthorized user gets their hands on the gun, like a kid coming across it.
Ok well, that doesn't seem very useful to me. The correct answer to the problem of kids is to lock up your guns. That is much more secure, particularly since something like this would only be effective if you didn't authorize you kids to use it, or remembered to remove their authorization when they were done at the range. Having them secured in a safe fixes the problem nicely. Likewise, that provides pretty good protection against theft.
So I really don't see what this will solve, and it will make things more expensive and complicated. It just doesn't strike me as very useful.