It's not so simple. There are a few points:
Are cars safer with backup cameras or without them?
If they are safer, do backup cameras prevent enough accidents/injuries to be made mandatory on all vehicles?
And these bits are specific to the OP and his 'Kids are run over in their own driveways. Backup cameras need to be mandatory.' point:
Do backup cameras prevent a significant number of kids being run over in their driveways?
If so, do backup cameras prevent enough kids being run over in their driveways to be made mandatory in all vehicles?
To the first point, I would say backup cameras probably make vehicles safer, with the caveat that it does nothing to the driver. Most accidents are the result of human error, and with any new safety feature, it has to be properly utilized in order to have any sort of effect. So yes, vehicles are safer, and drivers are given the avenue to being safer, provided they take advantage of it. (And given the number of people who don't use their mirrors or wear seatbelts, I'd say it's far from granted that a 'no-brainer' safety feature gets used all of the time.)
Second, I don't know. I don't have a strong opinion on whether or not backup cameras should be mandatory.
For the OP specific questions: Again, I think most of these specific questions (kids run over in their driveways) are because of carelessness or irresponsibility, things that are part of the driver's performance, and not the vehicle's capability. You might prevent some accidents, but not very many. That's not the 'wheelhouse' of a backup camera's application. You'll prevent many more two car collisions or pedestrian collisions (IE: jogger, neighbor walking their dog, etc.) than familial accidents. Since, if you're utilizing proper safety precautions already (knowing where your kids are in relation to your car) a backup camera isn't going to help you do that a great deal more, and if you're not doing that, the odds are slim that you'll use your backup camera properly, anyway. Either way, an accident isn't prevented. It either wasn't going to occur, or will occur anyway.
As for the second part, my answer's tied to the first. You might prevent a few freak occurrences, but it's an odd way to make your point. (That is, OP's assertion that because of the number of kids hurt in their driveways, backup cameras need to be mandatory.) There are stronger arguments for mandatory backup cameras, and if your whole argument is tied to a few outlying incidents, you'd have a hard time making your case. Additionally, you'd have a list of other, more pressing safety concerns you'd want to address before the backup camera one. (To draw a parallel. More parents teach their kids about stranger danger than fire safety, when fires are far more common. If backup cameras need to be mandatory because of the risk to kids in their own driveways, aren't there other things that need to be mandatory as well?)