You can't eliminate distracted drivers. (At least not until we all have self-driving cars.) I doubt that you can even reduce them significantly.
Being accelerated by 2 tons steel sucks, and helmets won't help you with that. However, once you're done accelerating, you'll travel some distance through the air and then you'll stop when your body comes into contact with some hard surface. It is at that point that the choice of hitting the hard surface with your skull vs. hitting it with a helmet starts to matter.
I've never met anyone who stopped cycling because of a helmet law. Have you?
Here's a conservative meta-review that tries to correct for several possible flaws, and it gives odds ratios of 0.43 to 0.58 for head injuries (that is, 42% to 57% reduction in head injuries when a helmet is worn):
http://www.quebecscience.qc.ca/files...eRuneElvik.pdf
Here's a very recent French study that finds 3x reduction in serious head injuries:
http://bmj-injuryprev.highwire.org/c....full.pdf+html
The root problem that makes it hazardous is the fact that you're trying to travel through busy streets without any body protection at speeds exceeding 20 mph, and your body isn't capable of withstanding crashes at 20 mph without injuries.
When faced with these risks, any rational community wouldn't even need helmet laws, because no rational, level-headed person would choose to get on a bike without one. You don't go sailing without a lifejacket, you don't go mountain climbing in winter without an ice axe, and you don't go riding a bicycle without a helmet. But, for some reason, bicyclists and motorcyclists aren't always as rational as, say, mountain climbers.
What makes you think that mandatory continuing education for everyone would have any effect on the problem?