Even if she prevailed, there's no guarantee that the problem would be fixed in a reasonable time frame, or at all.
I could change my name for ~$100, new passport and license costs, and an aggravating amount of paperwork.
I could convince the New York DMV to fix their record-keeping by... well, maybe if I convinced an influential politician to shame them for it. Maybe. I suppose I could devote years of my life to a public campaign about the issue. There's a chance that would work, though probably they'd just fix my file to shut me up and not fix the system.
If your point is that this is insane, then yes, yes it is. But changing your name is almost certainly easier than convincing a large, high-bureaucracy state DMV to solve the problems they cause.