The doctors were either not familiar with Georgia law:
If you’re past around 6 weeks pregnant, you may need to travel out of Georgia to get an abortion unless you qualify for an exception. Exceptions are very limited and include:
To save the pregnant person's life
To preserve the pregnant person's physical health
If the fetus is not expected to survive the pregnancy
If the pregnancy is a result of rape and/or incest.
The following video makes some good points about the logistics of getting one of these exceptions OK'ed. Apparently, it's not the doctor that decides when the life of the mother is at risk, at least not with legal impunity, as they were afraid to act.
Also, who says, "Yes, this was rape (or incest)" and grants the right to the procedure? Who do we suppose the states like Georgia that criminalize abortion (and now even a D and C) would empanel to make such judgments?
At 1:50, (the video runs 15 min 41 sec) Lawrence O'Donnell begins discussing the case of a girl who became pregnant at age 12 - apparently an actual case of a 12-year old raped by her stepfather that predated these new antiabortion laws, but in this context, it is a hypothetical 12-year old post-Dobbs in a state like Georgia - and why those legal exceptions in writing might not be enough to translate to action when action is needed:
The point is that these Christian theocrats are probably not really interested in any exception or the life of the mother. Individuals might vary, but collectively, when decisions are made, one shouldn't expect many if any compassionate exceptions. Notice that we haven't heard any expression of concern or regret coming from the right regarding the needless death in Georgia. Nobody in charge in these states with severe abortion restrictions seems to care that these laws are killing people. We haven't heard anything about reforming them from the people who created them and their allies.
In my opinion, O'Donnell is one of the better political commentators on mainstream media. He did another show on the story in the OP I believe the day before the video above outlining how George W. Bush and Trump filled the Court with theocrats, and now that they "got what they wanted," we see the fallout (19 min 12 sec):
I guess you shouldn't ride in a car in Georgia while pregnant, because if you get in an accident, the doctors won't be able to treat you.
A girl or woman shouldn't live in Georgia or any of the other theocratic states if she is fertile. It looks like there are 14 states with a total abortion ban (ID, ND, SD, OK, TX, MO, AK, LA, IN, KY, WV, TN, AL, MS) and 4 more where it's 6 weeks (IA, SC, GA, FL).
It looks like the so-called safe haven of NC where the girl in the OP was trying to get to is only slightly better:
"As of July 1, 2023, abortion in North Carolina is currently illegal after 12 weeks of pregnancy. In the case of rape or incest, abortion is legal through the 20th week of pregnancy."