At this time, driving is a privilege, and is subject to you following the rules while doing so. Much like signing your ticket, failure to do so is a jail-able offense. It's only for convenience that the cops let you sign something or only require ID while you drive. Otherwise they'd haul everyone to jail who broke the speed limit.
I can't say the cop overreacted, because he was really doing his job. The only point of discretion he had was pulling her over. After he did, he had to either do his job, or say, "Well, you win lady. No ticket for you. Bye!" and that's not going to happen...
I'm reminded of something I wrote earlier.
It was a surprise to learn that "Don't mess with Texas" wasn't a warning to oppressors to leave them alone, but a warning that the state is set in its ways and may fine you $10,000 for littering. To me, there isn't much difference between oppressively restrictive and oppressively polite.