"What's more, if you are thrown over the handlebars, you are FLUNG to the ground rather than just falling to the ground."
No, you really aren't (other than in some very weird scenarios involving cleated shoes that aren't relevant to children or casual riders).
Get a pen and paper and your O Level physics are look at what's going on. Going over the handlebars is essentially a case of the bike stopping and you carrying straight on (hence my caveat about cleats: if you're fastened to the bike you might rotate about the front axle, and the physics are a bit different).
Imagine you're cycling along at five metres per second. At the point the bike stops, the horizontal component of your velocity is 5 m/s and the vertical component is zero. Over the next second, you accelerate towards the ground at 10m/s/s. Your head is falling from about 1.5m up, so you hit the ground at 5.5 m/s, vertically. That's precisely the same speed you'd hit it at had you fallen over whilst walking. That energy is given up instantly, and is what will give you a headache. Horizontally, you were doing 5 m/s, which you're still doing 0.5s later when you hit the ground, and you will slide rapidly to a stop. Provided you don't hit anything, that energy dissipates relatively safely.
The risk with going over the handlebars is not head injuries. The risk is a broken neck, which is fearsomely common. That's because your head rapidly slowing from 5 m/s to zero is safe-ish for your head, but unfortunately unless you're Posh Spice your body weighs more than your head and your neck is therefore doing the job of a towing hitch. The faster you slow down (ie, the more rapidly your head decelerates) the harder your neck has to work, and in motorcycle land a lot of work goes into making the shells of helmets really tough so that the accident can be made to last as long as possible to reduce that load. That's not practical for most cycling purposes, so the helmets are foam with a cosmetic cover. That rapidly degrades in an accident, and may bring your head to a halt horizontally faster than would otherwise have happened. Thus it spares you a headache, but trades a sore neck for a broken neck: not a good deal.
A hideous number of cyclists die on the British roads, most of them from impact with cars (which is what kills them) not from the impact with the road (that doesn't). Post hoc "they would have lived had they been wearing a helmet" claims are impossible to validate, but the Dutch experience does not show a rise in the number of deaths relative the number of accidents. There is some evidence that there is also, in high helmet use countries, a rise in the number of serious cervical injuries. Pixie may have headaches; she isn't, by implication, in a wheelchair.
The CTC are not anarchist nutters rolling over the pavements on natty fixies. They're a sober, respectable organisation of cyclists, many of whom tool around on nice sensible Dawes Galaxies. Their view is that the evidence is equivocal at best, and that counts for a lot. There are some styles of cycling where helmets are vital (serious Downhill, for example), but there the helmets are much closer to motorcycle, or at least climbing, practice and people are also wearing body armour. For road bikes? Nah.
And for my big bike accident, had I not been wearing a woolly hat I'd have hat hypothermia to add to my woes. 