He sounds cautious, but I think the argument that 'studies show..' regarding motorists being more careful around cyclists is grasping at straws.
Of drivers on this forum do you eyeball all cyclists, pay attention to whether they have a helmet on and decide to drive more carefully?
Apparently the best helmets are mountain bike ones that cover the back of the head. Cycle helmets are extremely light and not constricting - give it a try!
Maybe he feels like a kid with one on, or has social anxiety about wearing any kind of head wear.
I sympathise there, having bought many hats over the years that look great, but I'm shy to wear. :)
If he wears a helmet he can cycle a little faster and get more exercise, so win-win.