Plus I doubt Kirk or even Spock would condemn them to death if they had known!
Indeed, they argue that the whole point of the exercise is to prevent Christopher's important son from being unborn. Supposedly, there will be no son unless the starship goes back to the past, again supposedly because Captain Christopher will "die in the crash"/be abducted to outer space before siring that kid. So the goal there is for the "second"
Enterprise to stop the "first" starship from taking away the pilot, or at least to return the pilot there immediately after the abduction. There is little need to "restore history" as such - what counts is allowing Christopher to live on.
So what might
really be going on is that the "second"
Enterprise, safely in outer space, beams Christopher into the cockpit at the same split second when the "first" one, struggling inside the atmosphere and blind to the presence of her alternate self, beams him out of there. So there's no "superimposing one man upon another", just the filling of a sudden cavity.
Once beamed back, Christopher is confused long enough for the first starship to escape (with the earlier Christopher on board). And the heroes have already taken care of the wing camera data on the previous iteration of the loop. So everything is fine: J.S. Christopher will be born, and USAF will have no material on the UFO. Assuming, that is, that the later Christopher was rendered amnesiac when beamed into the cockpit...
...And that part is probably within the capabilities of our heroes. Only Christopher says no to induced amnesia; McCoy actually thinks it might work. We don't see him wielding the necessary hypospray, but perhaps we just missed it when we blinked? Or then he slipped something into the chicken soup (or whatever Christopher ate - he, too, must have been fed somehow). Or then there is an "AMNESIA" button in the transporter console.
That the Sergeant is also beamed back is completely unnecessary from both the POV of his future and the POV of keeping the UFO secret, though. Kirk just needs to get rid of him somehow - so the "beamed down to his death" thing still very much applies.
Timo Saloniemi