GUMS:
I am having trouble understanding why over 40 years later we would have problems doing an in-flight alignment if all the nuggets did was turn off the inertial system(s).
Although I do not (at all) know the specific aircraft systems I have dealt with similar "allignenment/startup" cases in other fields.
My guess would be that the initial ground alligenment not only sets current lat/long/height/heading (and others) but may also fine tune
sensor calibration.
This last part is much easier to do when all inputs are static where inputs can be averaged over many samples.
While it woould be possible to save ( at least partially valid) values over a power cycle/reboot it would require extra code and HW as well as create yet another path to verify/certify, and probably was not high on the requirements list.