Why are third trimester abortions controversial?

[Skill and Chance]

IMO, the life/severe health risk to the mother exception arises from the right to self-defense.  It can be legal to kill an adult in self-defense under certain specific circumstances when one is at risk of death or great bodily harm.  Of course, this can apply to an intentional, premeditated attack, but the right to self-defense can also to the same sort of attack by e.g. an insane person who has no idea what they are doing.  A life-threatening pregnancy is analogous to the last scenario.  Because it can be legal to kill an adult in self-defense, the stage of development should be irrelevant. 

[Skill and Chance]

There is a deep seated repulsion at the killing of babies.  In ancient history this could at times be overcome by appeals to the needs of the community. Today it is more often argued for by appeals to needs which are (at least on the face of it) more personal in nature. But many people are hard to convince.

Sadly, I'm not sure this is correct. 

Even setting aside abortion, anti-infanticide cultures were few and far between over human history. Off the top of my head, places that consistently condemned it for at least ~1000 years: ancient Egyptians, ancient Israel, Christianized Europe from Roman Emperor Valentinian on, and Arabia after the Islamic conquest.  Ancient China outlawed it briefly, but the overall trend was toward it becoming more and more socially acceptable over time there.

Consistent opposition to pre-birth abortion since ancient times is basically just Christians, Muslims, and a few Greco-Roman stoics here and there (they were in a deep minority). 

[Skill and Chance]

Because this is a baby, not a clump of cells.

You had more than enough time to abort.  Should be limited to the first trimester.

Do you honestly believe that women who get abortions in the 3rd trimester are doing so electively?

Not officially, but I think there's a lot of "we'll do whatever you want and make the paperwork say what it needs to say" if you dig under the hood.