IMO: the idea that "this kind of argument exposes deeper asymmetries..." is itself fallacious for the same reason: it presupposes that a person's morality answers to logic.

Were morality a logical system, then yes, finding apparent contradictions would seem to invalidate it. But somehow that's backwards. At some level moral intuitions can't be wrong: they're moral intuitions, not logic. They obey different rules; they operate at the level of emotion, safety, and power. A person basically cannot be convinced with logic to no longer care about the safety of someone/something that they care about the safety of. Even if they submit to an argument of that form, they're doing it because they're conceding power to the arguer, not because they've changed their mind (although they may actually say that they changed their opinion as part of their concession).

This isn't cut-and-dry; I think I have seen people genuinely change their moral stances on something from a logical argument. But I suspect that it's incredibly rare, and when it happens it feels genuinely surprising and bizarre. Most of the time when it seems like it's happening, there's actually something else going on. A common one is a person changing their professed moral stance because they realize they win some social cachet for doing so. But that's a switch at the level of power, not morality.

Anyway it's easy to claim to hold a moral stance when it takes very little investment to do so. To identify a person's actual moral opinions you have to see how they act when pressure is put on them (for instance, do they resist someone trying to change their mind on an issue like the one in the OP?). People are incredibly good at extrapolating from a moral claim to its moral implications that affect them (if you claim that we should prioritize saving the lives of shrimp, what else does that argument justify? And what things that I care about does that argument then invalidate? Can I still justify spending money on the things I care about in a world where I'm supposed to spend it on saving animals?), and they will treat an argument as a threat if it seems to imply things that would upset their personal morality.

The sorts of arguments that do regularly change a person's opinion on the level of moral intuitions are of the form:

* information that you didn't notice how you were hurting/failing to help someone

* or, information that you thought you were helping or avoiding hurting someone, but you were wrong.

* corrective actions like shame from someone they respect or depend on ("you hurt this person and you're wrong to not care")

* other one-on-one emotional actions, like a person genuinely apologizing, or acting selfless towards you, or asserting a boundary

(Granted, this stance seems to invalidate the entire subject of ethics. And it kinda does: what I'm describing is phenomological, not ethical; I'm claiming that this is how people actually work, even if you would like them to follow ethics. It seems like ethics is what you get when you try to extend ground-level moralities to an institutional level. when you abstract morality from individuals to collectives, you have to distill it into actual rules that obey some internal logic, and that's where ethics comes in.)