you'd think they could devote some of their time to have someone simply set all the clocks on all the hardware for the time of that night's transition

This requires resources (time, money, and effort.) A lot of federal agencies, NASA included, don't have a whole lot of resources so they have to prioritize work for the most bang for the buck. This activity probably hasn't made the cut.

or point the software at an NTP server and set that to the time it transitions.

First RFC on NTP: 18 April 1981. First space shuttle flight: 12 April 1981. NTP wasn't invented (or at least standardized) when the shuttle was designed or even built. I'm guessing it isn't implemented on the shuttle.