If anything's it's the TOS feature films that introduce a continuity error with respect to McCoy's knowledge of alien anatomy. During the TOS series McCoy was able to perform complex heart surgery on Spock's father Sarek; and have more than enough knowledge to diagnose and treat Spock on a number of occasions.It will be continuity error if the Enterprise has a lot of rubber face regulars in Pike era. When the Enterprise had more humans than rubber faces in Kirk era (TOS). Also, it is also a canon violation too, as even McCoy still didn't understand too much about Vulcan and Klingon health, years after the event of this series.
So either McCoy was a stupid doctor in TOS, because he was surprised about Spock's blood color, confused on how to help a Klingon in Star Trek VI, or humans - Aliens interaction should be more limited in SNW era. Or you ask a reboot that change everything. If that what happen, a reboot, then the series should continue to TOS era again, and erase the old TOS canon.
I don't mind a reboot, but what about the other?
In TOS S2 - "The Trouble With Tribbles"; McCoy immediately recognized with one tricorder scan, that there was a Klingon agent surgically altered to appear human on Space Station K7.
For increased dramatic effect during the TOS feature film era, writers seem to like to make McCoy less knowledgeable on anything but human anatomy.
As far as the amount of alien integration into human crewed starships, once TAS aired, You saw many more alien crew members aboard the USS Enterprise because it didn't cost more money to draw aliens as compared to the cost of extra makeup for aliens on a live show.
Bottom line: No. Seeing more alien crew members on Christopher Pike's Enterprise, or having a CMO that actually knows a lot about alien physiology will not in any way be a continuity error.