Revealing mistake: Thornhill & the "Professor" have a chat on the airport tarmack, in which it's revealed to Thornhill that Eve is a secret agent and in danger. At that moment the camera dollies in on Thornhill for dramatic effect. As it does, the distant airplane in the background gets closer very nearly as fast as does actor Cary Grant, revealing that the background in this scene is, in fact, a rear projection on a screen just beyond the actor. (01:40:15)
Revealing mistake: Right after Eve "shoots" Roger and meets up with him and The Professor, some of the trees wobble as the ambulance drives past them. (01:46:20)
Revealing mistake: In the shooting scene in the Mount Rushmore cafe, a boy in the background puts his fingers in his ears, because he knows the gun is about to be shot.
Revealing mistake: When Thornhill is going up the driveway of the house near Mount Rushmore, there are shadows, even though it is night-time! These shadows are far too strong to be moon shadows and there is no source of artificial light that could be making them.
Revealing mistake: While in the police car, immediately after the driver hangs up the phone, he makes a sharp turn to the left (screen right). The driver and Cary Grant move in perfect sync with the rear projection; however, the policeman seated next to Grant completely misses his cue, remaining rigid and unmoving. Grant is then forced to push him aside and physically nudge him.
Revealing mistake: When Thornhill's cab arrives at the UN there's a couple behind him, a rear projection. Later on, when Valerian's car arrives at the UN the same projection is used, with the couple a mere two meters away from where they were before.
Revealing mistake: Just before Roger Thornhill exits Eve Kendall's train compartment, they are kissing and have their hands cradling each other's heads. When Roger turns his back to the camera and Eve's hands are behind his head you can see a tan line where Eva Marie Saint's wedding ring usually is. Her character, Eve, is single.
Answer: More than likely, they felt that Roger would be dead and they would not be found out. The fact that he survives their DUI plot and returns to the house with the police only serves to makes him look more suspicious and guilty. It's to move the plot along, nothing more.
ChiChi
The bigger plot hole is, if Van Dam really believes Roger is Kaplan, why would he think that Roger would bring the police and go through the trouble of preparing "Mrs. Kaplan" to make the police think he's crazy? If Roger really was a spy, he doesn't need help from the police and would have just disappeared instead of retracing his steps. So if Van Dam anticipated the actions taken by Roger, he must believe at some level that Roger is telling the truth and would have looked deeper into it.