Livre d'Or

in strongly and accurately portraying the Catholic Church it is also driving home some of the things I very much don't like about it
Yes. Part of the reason that I found the religion in aCfL plausible is that it evoked a very similar revulsion to the one I have to certain aspects of real Christianity. But I really liked the way that it presents sympathetic and sincere characters as accepting such principles and trying to live by them.

the emphasis on humankind's fallen nature
That's definitely one of the things that bothers me about both real and fictional Christianity. You possibly have more right to take issue with that attitude than I have, but anyway.

the triage scene near the end
To me that was a really amazing piece of writing. I really liked the way that the story got right inside the heads of both the priest and the doctor. It's not obvious that euthanasia is wrong, but nor is it obvious that euthanasia is right. I loved the way that Miller presents two honest, decent people coming from different perspectives to a morally impossible situation; I wasn't ready to condemn either stance.

I've always found the first novella a lot stronger than the last two
I liked the first a lot, thought the middle one was kind of ok but not really special, and the third was absolutely amazing. The first novella simply wouldn't stand alone. And it's the final few chapters IMO that make aCfL an outstanding rather than just a decent book.

I strongly advise against reading St. Leibowitz and the Wild Horse Woman
Thanks for that tip. I wasn't particularly planning to, since I really don't feel that aCfL is begging a sequel. But if it's set in the middle rather than after the end then I'm even less interested to read it, I think.