Well, no - if they were more common, it would lead to short-term problems, such as more people being kicked in the face, and people landing awkwardly after poor execution of the kick.
Ultimately, all sports are dangerous, to a greater or lesser degree. Every contact sport will have career-ending (possibly lifelong mobility-threatening) injuries, even deaths on the field. The question is whether the risk is acceptable. Now that football (soccer) balls are so much lighter, as has been pointed out, I think the risk of brain injury is much lower than it used to be - but more research is needed. If that research went on to show that, say, 10% of regular soccer players ended up with brain damage, to me (and I suspect to most people) that would be an unacceptable level of risk to take. The actual number will almost certainly be orders of magnitude less than that, because otherwise we would already have an obvious problem - hundreds of millions of people play soccer regularly for many years of their life, if there were a serious problem it would have come to light by now. Even if the problem were restricted to professionals (plausible, as they play and train harder and much more regularly than anyone else), I think we would be able to quote more than the one example of Jeff Astle (my apologies if there are other well-known cases I am missing).
Now, with gridiron (and boxing), the impacts are much heavier and more frequent. So there is inherently a higher risk of this sort of thing. It may well be that that risk is unacceptable. So far, society at large has mostly deemed it not to be. And so the NFL etc continues.
Back to the OP - such a rule change would significantly change the game - I’m not sure if it would “damage” it. Players would soon become (more) proficient at chesting the ball down and just avoiding their head altogether. It would make for strange play if a long ball was tried, as commonly two players from opposing teams leap up to head the ball - that would disappear, they would either have to let the ball bounce (likely making defending more difficuly - as a defender you are taught not to let it bounce if you can avoid it) or try to make contact with a shoulder, or something. But that would lead to the ball bouncing off shoulders into heads, and a lot of marginal calls - the “handball” rule is already one of the most difficuly laws to police, I don’t think we want to add to that difficulty. The only way really to make sense of it is to stipulate “no over head height” (the ball, that is) - a commonly-used rule in indoor football. That would certainly change the game - except for Barcelona fans, of course :).