Now to my question....why would anyone buying or selling a firearm for legal purposes, object to a bill of sale?
To answer this can we first explore why the seller wants a bill of sale? In my experience the seller is concerned about events following the sale, for example the gun is recovered at a crime scene and traced back to him. In this situation the seller is afraid that if he can not prove to the police who they sold the gun to that they will be subjected to an uncomfortable and/or expensive experience. The bill of sale would seem to resolve this concern, but it does not.
Consider the worst case scenario, and I've spoken to someone that this happened to. You sell a gun to someone and sometime later the FBI knocks on your door and tells you that a gun that you owned was recovered at the scene of a crime. For simplicity let's just assume a domestic mass shooting. The officer is knocking at your door. When you open the door he will either place you under arrest if there is credible evidence that you committed the murders or he will ask if you own the gun. You weren't involved, so all that's happening is a friendly, if nerve-wracking, conversation. The conversation goes one of two ways, in both cases the seller tells the truth.
If I was the seller I say that I have owned such a gun, if I remember that specific gun I say that I did own it. He asks if I still have it and I reply that I sold it. He asks who I sold it to and if I remember I tell him. I'm not as vigorously anti-government as some, I want the killer to be caught, so I'm sincerely helpful. Maybe he asks if I knew that I'd sold it to a felon, to which I respond that I always ask if they are a felon or otherwise prohibited from purchasing or owning a firearm and then I check their ID to ensure that they are a resident of NC and if it was a pistol I check their PPP or CHP to their drivers license and confirm their age. It's also possible that I sold it to someone from another state, in which case I shipped it to an FFL who would have delivered it to the buyer. Even if I can't tell him who I sold it to I tell him that I always follow that process. The officer may choose not to believe me, but he has no evidence that I have broken the law. The conversation ends amicably.
The scenario plays out almost exactly the same if the seller has a bill of sale. To the officer the bill of sale is just a piece of paper with some writing on it, it is not conclusive evidence that the sale actually took place, that the buyer is the name on the piece of paper, or that the seller attempted to determine if the buyer was prohibited and that the seller made a reasonable effort to verify the identity of the buyer. It is a piece of paper that the seller could have created 10 minutes ago, there is no way for the officer to know if it is what the seller claims. The lawful seller says exactly what I said above, assuming that he uses a similar process, and provides the bill of sale. As above, the officer has no evidence that the seller has broken the law and the conversation ends amicably.
This is the worst case scenario, and it plays out the same with or without the bill of sale, which brings us back to your question. Many buyers are offended by government overreach and the actions of citizens that grant to the government more power than they believe wise. These aren't criminals afraid that their illegal activities will be discovered, they are law abiding citizens sincerely concerned about the future of our society as they watch it become more Orwellian with each passing year. The extra-legal step of preparing a bill of sale and doing things like taking pictures of the buyer's ID or the license plate of his car offends these buyers because they serve no legal purpose, they imply that the seller is actively seeking to be an agent of the government (ie capturing data as if they were an FFL required to collect and maintain such information), and they expose the buyer to additional risk of crimes like identity theft as has been discussed above.
A great many sellers that want a bill of sale believe that the piece of paper protects them, and they believe that with a religious fervor with few supporting facts. A great many freedom loving buyers only want to do what the law requires and are offended by the inability or unwillingness of the seller to understand that what they are asking for doesn't accomplish what they think it accomplishes. Put it all on the internet and you might as well ask if 9mm or 45acp is a better man-stopper or if a 1991, Glock or CZ is the best handgun.
BTW, going back to the guy I spoke to, the conversation went exactly like the first above. He had no bill of sale, was as helpful as he was able, and had only a single conversation with the officer at his home. Nonetheless the experience traumatized him and now he requires a bill of sale and a copy of the drivers license...I can't explain it.
It's been a long day, and I apologize for the long post. It is not intended to cover every possible scenario, but if you put some thought into why you think a seller would want a bill of sale I think that you'll realize that the document really has no value in any scenario.