Elder care issues are difficult. Judging from your post, the decision lies with your FIL. The concerns of the children are not substantiated. The physical issues described are not relevant to that decision. Any estimate of "bad judgment" is at best only a layman's judgment absent any actual evidence.
Keeping a loaded pistol by the bed is no big deal. It is routine for countless millions of people, both those who use guns routinely and those who seldom if ever fire a shot. Some even (perish the thought) keep a revolver or pistol under the pillow. I'm not aware of any reports of such revolvers or pistols cocking and firing themselves while the owner slept. On the whole, it is a illegitimate concern.
As in any other discussion of elder care, the concerns and interest of the individual are not to be considered in isolation. In this case, your MIL is rightly involved and must be considered. This is for the simple reason that they are husband and wife. Regardless of what anyone else thinks in the matter and absent any extreme reason to the contrary, the decision is theirs alone to make. If your MIL does not like guns, that is her choice. It will enter into their decision.
If mobility up and down stairs, etc. is becoming a problem, this needs to be addressed promptly. Loss of mobility due to injury, especially broken hips, is very much a factor in health decline among seniors. It can require a expenditure of money, etc. But the cost are far less in money and life experience than the cost of hospitalization, therapy and possible subsequent institutionalization.
Again, judging from your post, your FIL is about at the point where his ability to safely drive a car in normal everyday traffic needs to be seriously evaluated. On that basis alone the decision should be made exclusive of emotion or philosophy. His independence is one thing. But the risk he can pose to himself and others is entirely a larger consideration. As much as everyone talks about the accident rate among teenagers, the rate, etc., among elders is just as significant.
Elder care issues are difficult. Much of that difficulty is due to the emotions involved. Making such decisions for someone with whom you are not related by birth, marriage, etc. is one thing. Making such decisions for a family member, etc., is very much another thing entirely. Some will urge nothing except the minimum be done out of concern for personal freedom, etc. This is at best misguided. Some will urge nothing be done out of personal concern that they might have to face the same problem with a consequent loss of their own personal freedom/prerogative. This is not without merit. Again judging from your post, your FIL and MIL are capable of dealing with this matter to their satisfaction. If in the future your father begins to fail mentally, the decision will fall to your MIL. That is how it should be. They are married. Everyone else is at best spectators.
Speaking personally, my father drove his vehicles without problems until he was 83 years old. At that point he was living alone my mother having predeceased him. My brother and I decided that it was best that he no longer be driving. The keys remained on the dining room table. My brother is a very capable mechanic. He disabled the car and truck. It worked in our situation. My father's COLT Det. Special remained in his bedside table until the day of his death. Again, it worked in our situation. HTH. Sincerely. brucev.