Regarding the annulment issue:
I think there is a big difference between people who are concerned that the annulment process is subject to
abuse in some way, and people who actually struggle to accept the theology behind it. The two issues tend to get confused a lot, from what I have seen. I think most Catholics accept the theology, but many find its application to be in error, or “corrupt” in some way.
And BTW, shady annulments are actually NOT completely new. Many of the royal marriage annulments in the medieval/Renaissance era were based on rather flimsy rationales, mostly to do with degrees of consanguinity or the interpretations of prior betrothals as prior bonds. Hence why many Anglicans state that the Church denying an annulment to Henry VIII was all about politics, not principle.
It doesn’t help that a lot of secular approaches to history blur the line between annulment and divorce; until I started to explore Catholcism, I myself was under the impression that Henry VIII wanted to
divorce Queen Isabella, and that the split with the Church was over whether divorce was acceptable.
I’ve also read accounts of King John of England (the one popularly termed the “worst King ever”) and his marriages, that state he “divorced” or “set aside” his first wife, Isabel of Gloucester, so that he could marry the young (and rich) Isabella of Angeloume. Now, the “real story” is pretty icky anyway, Isabella was very young at the time of the marriage, some think only 9 or 10 years old, yet it’s pretty obvious it was consummated.

But he didn’t actually divorce his first queen, he claimed his first marriage was invalid because Isabel was his cousin, and he didn’t get a proper dispensation to marry her.
Apparently, John did actually love her, he had many mistresses while married to his first wife, but none after he married Isabella (there are rumors he may have strayed once in a while, but he had no public, “official” mistresses).
Now, was this annulment questionable? Did he put pressure on the Church to grant him the annulment? Most likely, yes. But that doesn’t make the whole theology of annulment invalid. One could also argue that the real corruption was not in the granting of annulments, but in allowing royals to marry in shady circumstances that left the possibility of annulment open to them.
I’ve also noticed that many Catholics (as well as non-Catholics) who oppose capital punishment do so chiefly out of concern that the process is subject to abuse, not because they think capital punishment is inherently evil – and actually, I understand that the Church does NOT teach that anyway.
It seems many people who oppose the death penalty do so because of a perception that the legal process in general is subject to error, bias, or even outright corruption, but if a theoretical all-knowing perfect judge was to mete out the punishment, this would be legitimate (and indeed, there are many instances in the Bible where God Himself did strike people down).