Other mistake: The female journalist is called Samantha Coleman through the movie, but appears in the credits as Samantha Copeland.
Other mistake: McClane pulls out a 9mm Heckler and Koch MP5 submachine gun, aims it at the police chief and pulls the trigger. It's loaded with blanks but nobody except him knows that. The problem is that the police chief is surrounded by armed colleagues, and not one of them draws his gun in defence of his boss. In reality McClane would be riddled with bullets before he could pull that trigger. Not to mention an MP5 needs a blank firing adapter to use blanks (it fits on the muzzle via three lugs), none is fitted on the gun in the scene. As a blank firing prop gun it would use a hidden internal restrictor, but that wouldn't be feasible if it was meant to be able to switch between blanks and regular ammunition, as it's supposed to in the movie.
Other mistake: When the platform falls on top of McClane and he drops his gun, I don't see how it would land more than six feet away from him. One foot at the most. Unless of course it was made of rubber. (00:47:05)
Answer: Because the police were moments away from arriving at the scene. They needed as much time to get away without being tracked. Every moment counts. And a target rocking about a few hundred feet in the air doesn't seem to be such an "easy target".
XIII