There’s a couple threads here, so I’ll attempt to delineate my responses:
Regarding the football comparison: there isn’t generally speaking a right to play football, and I’d prefer if sports bodies were committed to ensuring the health of players of their sport, but no, I don’t think that sports bodies should be relied upon to enforce some arbitrary standard for safety in their sport, with the threat of banning football. The football example is interesting because smashing somebody in the head as part of a football game isn’t “really assault”. If you and I agree to play football, and during the course of the game we collide and I get injured, you didn’t assault me. The problem with sports bodies is that they have several incentives that don’t align with overall player long-term health, and the players typically (especially historically) don’t have a strong enough negotiating position to push for systematic change. So to your question: no, I don’t think we should expect sports industries to carry the weight of making their sports “safe”, I think society should agree on laws that restrict sports bodies behavior so that they’re forced to operate in ways that are safe. We don’t ask construction sites to please make worker conditions safe, we set up OSHA and equivalent orgs, with regulations to mandate safety.
Regarding the job that football associations have, that’s more important than long-term health, “profit” is the answer. The football organization isn’t a benevolent charity, they exist so as to ensure their own continued existence. They also don’t represent the players, nor the fans.
Switching gears to the NRA: the NRA has a similar job. They exist in order to perpetuate their lobbying influence and perpetuate their funding. They are not an accurate picture of the average American gun owner. They push for or against policies on the basis of continuing that power, more so than any intrinsic goal of “avoiding legal restrictions on their members”, except insofar as their members are “NRA large-scale donors”, which effectively amounts to “gun manufacturers”.
Regarding it being “their problem to solve”, you still haven’t addressed my critique of that premise. Gun ownership in America is a right, written into the foundation of US government. There’s a whole body of debate as to whether or not it should be a right, whether it should be written differently, etc, but at present, gun ownership in the US is written into the constitution at a similar level to freedom of speech or protest. We don’t tell free speech advocates to find ways to limit hate speech or we ban all free speech, why would we approach this any differently?