Maybe. I do not know.
However, what I do not know is why football helmets are not designed like baseball catcher helmets.
Baseball catcher helmets are in TWO pieces: A front piece and a back piece. These pieces are attached/connected with "springs" basically, such that any front impact limits how much the HEAD moves. If a ball hits the front of the helmet, the front piece of the HELMET moves backward, but not the HEAD.
In football, you would probably need stronger "springs" connecting the 2 pieces, but the principle would be the same: Limit as much as possible how much the head moves after impact.
The problem with football is that though the HEAD is protected by the helmet, the BRAIN is not ... when the head is hit, the brain rattles around inside the skull. Helmets should be designed to limit the HEAD moving, or limit the effect of head movement, so the brain rattles around in the skull with much less violence or force. A 2-piece helmet with these "springs" connecting them would not prevent the force of the brain rattling with a football hit, but it should limit it, or mitigate it.
These helmets would not protect against a side hit, but I see no reason why they would not help a lot with frontal and back hits.
Sorry, this isn't really correct.
Mechanism of injury as relates to internal organs is 100% about acceleration. Your catcher's helmet example doesn't apply because a catcher's head is never moving, therefore the catcher's brain is never moving. The only thing moving is the ball. When you get hit in the head with a ball, the mechanism of injury is completely different from a closed head injury sustained while you are in motion. Those helmets wouldn't help football players.
The rubber idea is crude. but there's some logic behind it. Preventing internal injuries (brain and other organs) from decelerative MOIs is all about reducing the rate of deceleration, i.e. absorbing the impact.
The best analog is a car. Back in the days when cars were basically two-ton blocks of steel a lot of people died from internal injuries because the car would collide with something and would, itself, absorb very little of the impact. The collision forces get transferred, instead, to the vehicle occupants. They would, per Mr. Newton, tend to remain in motion until they meet with something that stopped them (seat belt, dashboard, windshield, concrete, whatever). At that point, their internal organs would tend to remain in motion until something stopped
them - rib cage, cranium, etc. The impact forces are then terminally transferred to the organs, causing catastrophic internal injuries.
Fast forward to today and you see collision energy being absorbed by the vehicle's structure. This serves to reduce the impact forces which are transferred to the occupants and reduce internal injuries.
As relates to a football helmet, when your head hits something there would seem to be insufficient impact absorption within the helmet itself. The shell of the helmet has zero energy absorbing properties, so all it can do is transfer energy to what's inside it - some amount of padding and your brain. If we choose to consider that there are too many concussions among football players then the construction of the helmet has to change.
There are plastics available that are "soft touch" but which maintain their shape and are impact resistant. If you have a high-priced car, you probably have seen such materials. Fabricating helmet shells out of a material such as that might be a good place to start.