Yes, because the first thing post offices want it to deal with is firearms. That isn't their job, nor is it something like that is even remotely free from forgery and easy to enforce.
And again, how do you enforce that? Does the government have the ability to know every time some one sell's their used coffee table to a friend? Like it or not, a gun is the owner's private property. The government stepping in and saying "No! You have to make your own private sales under our watch!" is NOT going to fly well, and then, when does that control end? How far down the rabbit hole of the government intruding on your life do you want to go?
No one would agree to that, because that is just added expense and man hours for the dealers and police who have to deal with that paperwork, which will trickle down into fees for the buyer and seller. I'm not trying to be a hard ass (I agree that states should take a hard look at their current gun laws), but everyone talks about "MORE GUN CONTROL!" with little to know idea of what can or cannot/should be done.
Yes, some states will reinstate a state felon's right to own a gun (which is freaking stupid, I'll add), but the VAST MAJORITY of felons will NEVER own a gun legally again. And there is absolutely no reinstatement given to federal felons. This is a point where I was talking about earlier - things that can REALISTICALLY be done: get rid of the ability that some states have to grant gun privileges to state felons is a great start.