NO, a higher licensing bar for human drivers will NOT solve the problem, it would only exacerbate it (and I'm ALL FOR setting a higher licensing bar for humans for other reasons).
The problem here is NOT the untrained driver -- it is the attention span and loss of context.
I've undergone extensive higher training levels and passed much higher licensing tests to get my Road Racing license.
I can tell you from direct experience of both that the requirements of high-performance driving are basically the same as the requirements to successfully drive out of an emergency situation: you must
1)have complete command of the vehicle,
2) understand the grip and power situation at all the wheels, AND
3) have a full situational awareness and understand A) all the threats and their relative damage potential (oncoming truck vs tree, vs ditch, vs grass), and B) all the potential escape routes and their potential to mitigate damage (can I fit through that narrowing gap, can I handbrake & back into that wall, do I have the grip to turn into that side road... ?).
Training will improve #1 a lot.
For #2, situational awareness, and #3, understanding the threats and escapes, there is no substitute for being alert and aware IN THE SITUATION AHEAD OF TIME.
When driving at the limit, either racing or in an emergency, even getting a few tenths of a second behind can mean big trouble.
When you are actively driving and engaged, you HAVE CURRENT AWARENESS of road, conditions, traffic, grip, etc. You at least have a chance to stay on top of it.
With autopilot, even with the skills of Lewis Hamilton, you are already so far behind as to be doomed. 60 mph=88 feet/sec. It'll be a minimum of two seconds from when the autopilot alarms before you can even begin to get the situation and the wheel in hand. You're now 50 yards downrange, if you haven't already hit something.
Even with skills tested to exceed the next random 10,000 drivers on the road, the potential for this situation to occur would terrify me.
I might use such a partial system in low-risk situations like slow traffic where its annoying and the energies involved are fender-bender level. Otherwise, no way. Human vigilance and context siwtching is just not that good.
I can't wait for fully-capable autodriving technology, but this is asking for trouble.
Quit cargo-culting technology. There is a big valley of death between assist technologies and full-time automation.