I rather think this is a bona fide plot point...
In "Space Seed", our heroes thought that the way to study the sleeping leader was to set his cryo-facility to thaw right there and then. If the facility were a removable pod, then the obvious choice would have been to move the pod with Khan still inside to the Enterprise
for closer study.
It is certainly possible that all the Augments were moved from their original chambers to these pods at some point. But McCoy thinks cryosleep is a lost art in the 2260s, so there shouldn't have been a ready supply of such things. Doesn't mean S31 couldn't have acquired such pods somehow - or even that the starship that found the Botany Bay
would have been unable to construct such pods on the spot.
In any case, apart from the faces of the heroes, these pods would be our very first thing that has been seen both in the new movies and the original continuity. Everything else in the movies (not counting things never seen in the original continuity and thus presenting no continuity problems anyway) is from an alternate timeline, arguably manufactured well after Nero stirred the timestream - up to and including the skyline of San Francisco!