Except when anyone ever watches the episode, they will not be thinking, “Oh, that was that time that Kirk used a different name,” but that it was simply before they chose the one they settled on. The episode, and all those after, didn’t want Kirk to have multiple middle names — just one. Just that the one they gave us changed...
You are being unreasonable. How many people do you know? What is the precentage of them that you are certain that you know the full names of?
How many people do you know that you are certain - instead of merely assuming - use their first name instead of using their middle name, if any?
How many people do you know who you are certain never changed their name in their life?
The other Tos episodes showed KIrk only using the name of James T. Kirk during those episodes. They never showed Kirk using only one name in his entire life. "Where No Man Has Gone Before" only showed Kirk using, or being described by, one name, James R. KirK, during the time of that episode. Where No Man Has Gone Before" naver said that Kirk never used another name in his life.
Taken together, "Where No Man Has Gone Before" and the other TOs episodes indicate that KIrk's name changed. Which is a very common and normal occurance.