Why doesn't Jimmy reveal himself to Bob in "After Twenty Years" by O. Henry?

The narrator's description of the as-yet-unidentified cop makes him look like a man who has been on the police force for many years—perhaps as long as twenty years. Bob's description of Jimmy to the cop (who he doesn't recognize as his old friend) makes Jimmy appear to be the kind of "staunch" man who would be devoted to his duty. Once Jimmy recognizes Bob as the man who is wanted by the Chicago police, he knows it is his duty to have Bob arrested. If Jimmy had been a different kind of man, he might have introduced himself to Bob and then told him to get lost. Jimmy couldn't do that. If he introduced himself, Jimmy would have had to make the arrest himself, and, as he says in his note,

Somehow I couldn't do it myself, so I went around and got a plainclothesman to do the job.

Jimmy could only see two choices: arrest Bob himself or have another cop arrest him. There were no other options for a man with Jimmy's character. If Jimmy identified himself and made the arrest, there would have been an emotional scene, with Bob pleading for mercy on the basis of their old friendship. O. Henry didn't want to write that kind of scene.
It is interesting that Dashiell Hammett writes such a scene in the final chapter of The Maltese Falcon. Brigid O'Shaughnessy begs and pleads and uses all her sex appeal to persuade Sam Spade not to turn her over to the police for murdering his partner Miles Archer. Spade is adamant, though. He tells her in beautiful American vernacular:

I don't care who loves who I'm not going to play the sap for you. I won't walk in Thursby's and Christ knows who else's footsteps. You killed Miles and you're going over for it.