We played some teams that wore helmets with foam padding outside when I played high school football. You know, back in the early '60s. Duke wore them at about the same time. Needless to say, they didn't work quite as well …
The main problem then is, I bet, the same problem now: the weight of the helmets. The things have gotten prohibitively heavy and, from what I hear, are a bother to wear. More padding will make that worse. That's why our coaches didn't adopt the idea way back when. (Well, that and there being little evidence that extra padding helped.) High tech materials will probably make this difficulty go away.
But 33 is right. It's a violent sport and the physics is getting dangerous for people's brains. Back when I played, the biggest guy on my college team weighed 230. Collisions were not as big a problem as they are now. I confess that I don't know when or if this problem can be solved to the extent we'd all like.