It’s splitting hairs to argue that “assign” can’t be used to describe this process.
No word should be used to describe "this process" because it's not the process that's significant. If due to some sort of bureaucratic error - or maybe even medical error due to a DSD - someone's sex is recorded wrong initially, then that shouldn't be locked in irrevocably.
Using "sex assigned at birth" in a place where actual sex is important presupposes historical errors aren't possible. (I can't actually think of any situation where you'd want to mention "sex assigned at birth" rather than actual sex, except maybe in as part of an application to get those records corrected).