Well, Scott did do a check of the transporter equipment before beaming Kirk up, and nothing was apparently wrong. And Fisher, once he materialized, didn't seem to be any the worse for wear other than the dust on his uniform and the cut on his hand. It appeared to be a one-time only thing with no damage to the machinery. Why shouldn't he have beamed the captain up? And even if Scott had stayed in the transporter room, the duplication still would have happened, although the second Kirk would have been spotted and caught sooner than he in fact was.
It always amazed me that a ship the size of the Enterprise only seemed to have the one transporter room. But I don't recall any on-air references to other transporters on the ship, although they must have existed, for cargo loading/unloading and emergency evacuations if nothing else. In the episode "What are Little Girls Made Of?", Korby speaks of removing the android-making equipment from Exo III to the ship, but how would he ever have gotten it aboard? Certainly not with the transporter that we saw on the ship, and it couldn't have been loaded on a shuttlecraft either. There must at least have been cargo transporters.