Yeah, the issue with the Pro-Life movement is we're marred by the Anti-Choice movement, which is the moniker I've given to the f*****g idiots who do nothing but shame women and scream things like, "Keep your legs closed next time!".
That accomplishes nothing but degrade the woman involved. Then they go on further to hurt the movement by focusing on things like post-birth deaths. As you noted, those are not only so rare as to be more hindrance than help in terms of how it impacts perception of the matter, but it's arguable that in many cases it's not really a good cause. I 100% think any human being who can be resuscitated and live a normal, natural life deserves that, but at what point do our good intentions turn into torture for a being doomed to die anyways? The concept of 'mercy killing' is another ethical bucket of worms, but wholly different from acknowledging that someone is doomed and prolonging the suffering by staving off death is morally compromised.
As you noted, most people presumably don't want to get an abortion, and instead feel they have to do it for a myriad of external pressures. Unless we understand this, no progress on the matter will be made because we'll just be shifting the problems around, not solving them overall.
One thing I always try to bring up when discussing abortion is the issue of the future of the child; what good is it saving their life if we drag the parent(s) into economic disaster as well? Some folks simply can't afford another child, and for all the, "I took on a second job to cover the costs." stories there are just as many, "I couldn't bear to kill him/her, but it cost me my job and now I'm destitute." to match. To that end, I think we need to seriously revamp and improve our foster care systems. I can only support ending abortion if we ensure these kids are being put in systems that look out for their well-being as best as possible. Being Pro-Life, at its core after all, is about respecting the dignity that everyone deserves throughout their lifetime. Couples who truly can't raise a kid need to be given an outlet that protects that kid's right to life while relieving the financial strain.
(I really didn't mean to preach here, but this is about as succinctly as I can put my position on the matter.)