'Assigned at birth' is something that used to happen to the tiny number of babies that were born with a DSD that made their sex difficult to determine visually. The phrase frequently meant horrific genital surgery so it has traumatic associations for a tiny number of people whose medical condition was mismanaged.
That doesn't happen now because, as you say, there is DNA testing. Also surgeons tend not to do surgery on tiny infants these days unless it's strictly medically necessary.
Aside from that, nobody has ever been 'assigned at birth', the TRA just coopted the phrase because they don't give a shit about anyone else.