For road riding, are mountain bikes safer than road bikes?
I would say it's debatable and depends on the individual's preference. Mountain bikes (MTBs) obviously have more traction due to wider and more grippy tyre designs. However a road bike could hold a turn better due to lack of shoulder on the tyre. Speeds tend to be lower on a mountain bike due to added weight and those grippy tyres. On the other hand these slower speeds could be a disadvantage in road traffic. (for reference I ride both an MTB mainly off road and a hybrid with road tyres for paved trails and roads)

I would recommend looking at better/further safety equipment (knee pads, elbow pads, gloves etc) and/or training rather than a new bike. For one, It's cheaper, but it will also install confidence. 

Another option is to try with a rented mountain bike to see which your son prefers.