Before jumping into the tech forum, how big is the Yorktown anyway? 16 miles sounds very conservative - It sure blew my socks off!
About 82448 cubic kilometers in volume. The total surface of the sphere is about 9160 square kilometers. I.e. if the internal surface would be habitable (Bernal Sphere-type space colony), the living surface would be larger that the total surface of Corsica.
Well, it's Kelvinverse, not main timeline) Maybe here the nubmer of M-class planets are significantly lower)...In our reality. In Trek, it's explicitly different, with endless Class M worlds available - and Spock has to give an exceptional rationale for Yorktown to McCoy who clearly feels that Yorktown is an exception to the style of UFP colonization or SF base-building familiar to him. That, too, was a nice touch in the movie (very little of the "cabbagehead" exposition trope there that didn't have a good in-universe excuse!).