Who did see the gun discharge at the back of Lincoln's head? Not that I doubt Booth was the perpetrator. I doubt there were many (if any) witnesses to the "gun firing" event. If only Oswald had leapt onto Elm after the shots.
Probably no one saw Booth pull the trigger but there is no doubt that he did. That is the point. You can make a logical inference. Someone hears a gunshot and then sees a gun pointed at the person who has just been shot. What conclusion would a normal person reach? Maybe that the shot had just been fired from that weapon. Not that someone else shot Lincoln or that the object being held wasn't a gun but an object made of "wood." The issue here is whether there was a gunman in the 6th floor window as confirmed by Brennan and others. The answer is yes. Does that alone mean Oswald was the assassin? Of course not. But that is just the first link in the chain. The police find a rifle and bullet casings on the floor/window from which witnesses confirm the shots came from. The second link. That rifle belongs to Oswald an employee in the building. The bullet casings found by the window are from his rifle. More links. Oswald has no credible alibi for the moment of the assassination. He flees the building, has a bizarre political background. He is implicated in another murder less than an hour later. He resists arrests, lies to the police about his ownership of the rifle. Under the totality of circumstances and facts, the case is a slam dunk of guilt. Any counternarrative premised on Oswald being framed or blundering into guilt by, for example, lying about matters that would have assisted his cause by telling the truth are laughable defense attorney tactics to create false doubt regarding a client they know is stone cold guilty. It's absurd nonsense.