The Sap and the Heater: a Design Sketch

If you had a game with a wider range than basic military combat scenarios, you could create simple but interesting mechanics out who gets the drop on whom.

For example, you're playing your Chandler-styled 30's detective game and snooping an upscale hotel room crime scene before the police show up.
Unfortunately for you, the goons of your shadowy nemesis had the same idea and have just arrived outside the door.
Hearing the door creak, you could pull your .45 before they get around the door, maybe even shooting one of them through the door.
Here, the alert player is rewarded with scenario played his way, not according to a canned "thugs enter the room" cutscene. Or the goon steps into the room without seeing you, and so is still at your mercy if you get your gun trained on him first. He can try to draw but you'll always be able to react faster with a pistol already in hand. You are free to shoot him or try pumping him for some information before pistol whipping him/shooting him.

Say you didn't hear or see him, so he draws his gun on you and then tell you to put your hands up or disarm, if you were in the process of drawing. He drew first, now he has the upper hand.
Then you try a verbal feint ("look over there!") or persuasive argument, or a grappling move if you're close enough. Anything to get him to break eye contact or allow you to dive for cover--anything to break the "holdup" phase.
Disarming his weapon puts you back in charge; mutual disarmament might prompt a close-quarters fisticuffs. Only after these possible paths does the encounter becomes a more traditional pitched firefight.
Or yet again in place of a firefight, maybe you attempt a feint and are unsuccessful, meaning the goons disarm and sap you with a blackjack, leaving you to wake up as the police arrive (losing the potential clues at this location). This is a setback, but not a full stop/fail-retry game over.

The point of this design sketch is that every encounter need not instantly ramp up to a full firefight, allowing even a game mostly about shooting to have much more varied basic fighting. The game can still be nominally about shooting, but there is a lot of rich potential gameplay missed out on in most games in favor of always "cutting to the chase."

For reference, consider a clip from the original Miami Vice show. An actual expert shooter was used to act out an assassination, using a more realistic version of the sorts of balance of power-play just described: