After all, the name change to "Christopher Pike" happened very shortly before shooting began. For most of the development process, the captain was named Robert April (which eventually got used as the name of a separate character in the animated series)...
My understanding is that the April character in TAS was intended to be the original captain of the Enterprise, even before Pike. So the 1701 was launched, captained by April, then Pike, then Kirk.
ETA: Just checked the episode "The Counter-Clock Incident" and Kirk's opening log states that April was the first captain of the 1701.