There's quite a few problems with putting this limit here.
Firstly, we find that in practice these situational exceptions don't work, because they sometimes require the woman to go to court, and that can easily take longer than the entire length of a pregnancy. So, in effect, there is no exception, and these women will sometimes be forced to birth at tremendous risk to both their mental health and personal safety (given that we live in a country that thinks rapists should be eligible for child custody). On the rare occasion a woman ever presents for an elective abortion this late, due to something like rape or incest, it is usually because she's been being abused and held captive and couldn't get out sooner, and I don't think we have a right to abuse these women even further. But this is very rare. I'll tell you what's much more common...
Most fatal and otherwise very serious fetal defects cannot be preemptively detected by doctors until around 20 weeks. That is the main reason abortions occur after this point. Women who abort at, say, 21 weeks, are usually doing it for very serious medical reasons.j
Often, such health-related abortions aren't considered "emergency" situations. The woman might be safe, as of this particular moment in time. But then her choices are to just wait to go into preterm labor, hoping the deceased fetus doesn't start decomposing and poisoning her blood stream, or that she may wind up birthing a mortally defective baby who then spends a short few moments suffering before ultimately dying.
Basically, this is refusal to treat a woman until she is actively dying. Again, putting her life at risk for absolutely no reason. We could have treated her for what is a valid and serious medical problem when she was still healthy, and had a better chance of surviving the operation. Forcing doctors to wait until she's dying will basically create a situation like what we see in Ireland, where women sometimes die needlessly because doctors couldn't intervene until she was already succumbing to sepsis. That is insane. This is like if your doctors refused to treat you for cancer until your organs started failing.
Even if the woman manages to maintain her health, we are then telling her she must be forced to carry around when she knows is a fatal pregnancy for another 10 or 20 weeks, only to have a still birth. That is unbelievably cruel, along with being incredibly risky.
It makes no sense to force women into these situation. It puts their lives at risk, and sacrifices their mental health at the altar of what are ultimately symbolic "gains." Not to mention how cruel it is to mortally deformed babies who may experience nothing of life but a brief agony before they expire.
Further, it's unconsitutional. SCOTUS has ruled that the law has no right to interfere with a woman's right to abort a non-viable fetus. 20 weeks is definitely not viable. No fetus has ever survived under 22 weeks in all of medical history. But, just as it was in the 70's, the limit of when a baby has any reasonable chance of survival is still 24 weeks. I don't think that will change until we invent a womb substitute. 20-week fetuses have no lungs, no brain, and sometimes no skin. They are definitively incapable of life, and therefore this is an obvious violation of the Roe v. Wade ruling and womens' constitutional rights.