Yup - I mean, perhaps he became Chief Medical Officer at the start of the Enterprise's five year mission under Kirk. Though it's only established in Star Trek (2009), I thought the prime universe crew could have come together in a very similar way. The entire bridge crew, for all we know, could have potentially served under Pike (perhaps in different capacities) towards the end and remained there when Kirk took over as her captain.
Point is, in the prime universe, we don't really know what brought Kirk and his crew together or when they tours on the Enterprise began. Lots of room to explore and establish that, in my opinion.