"The Cage" was 2254, so if the date information is correct, DSC would be about a year later. Or maybe TOS was just a 1960 television production attempting to approximate the future but being limited in its ability to depict future technology. Why do we have to take every detail literally? Roddenberry himself essentially treated the show as a dramatization after the fact, like Dragnet. Or rather, a dramatization before the fact. I think it's best to assume that there's an underlying Platonic Trek universe that the various shows are merely interpreting, filtered through their own technological abilities and artistic styles.