The police have no duty to protect or deter crime, they investigate and attempt to prevent crime through deterrence. The supreme court of the united states or SCOTUS has ruled as such.
I am a part of a community and therefore responsible, in part, for it's condition and safety. I am responsible for the security of myself and my property. Part of that security is deterrence, also known as a gun.
Likewise, if I were a criminal and wished to target an armed person or persons I would use surprise as well. The fact of the matter is that the vast majority of criminals want a soft target and the presence of a gun has been stated as a reason to avoid that target. Think of it this way: If 50% of criminals who wanted to rob me wouldn't do so because I had a gun, that's 50% fewer criminals who will attempt it. If my OC firearm prevents half the robbery attempts on me, then it works better than a CC gun which the full 100% of criminals didn't know was present. A CC gun prevents no crime or attempted crimes until it is drawn or shown. An OC gun presents itself immediately and says this wallet isn't worth the potential loss of your criminal life. The other 50% of criminals didn't care about the gun, so it doesn't matter in those instances.
If a criminal is planning to the point of specifically eliminating me because he views me as a threat, then he's going to have done some research on me and I must have something he really wants. At that point there isn't anything I could do to prevent it from happening.
We can play what if games all day, but the fact is OC can function as a deterrent while CC does not. I'll take the deterrent that is obvious, because it does encourage the criminal to pick another target. Kind of like a big intimidating dog helps deter home burglary, because nobody wants to get bit or have the dog wake up an armed home owner who then presents a lethal threat.