Interesting essay. I'm so accustomed to naturalist consequentialist philosophers these days that to read Paul's piece feels like landing on an alien planet. :) There's such a huge gap between how we understand ethics.
There are two ways to read what Paul is saying. First the less charitable account. I think the project of trying to ground ethics -- whether in naturalism or in religion or in anything else -- is confused. We care about what we care about. A lot of people, including myself, care about the suffering of other organisms, and we want to reduce that suffering. This isn't the same as the "ethics of care" that Paul tries to refute. I'm not saying that people should treat animals well in proportion to how much they care about animals. Rather, I'm saying that *I* care about animals, so I'm going to do what I darn well can to make sure everyone else respects animals to the degree I think they should. :) It's as simple as that -- there's nothing more to say about the ethical underpinnings of animal welfare.
Now, the more charitable interpretation of Paul's essay is to read it not as talking about anything metaphysical -- not as talking about absolute ethics built into the fabric of the universe -- but rather, just as talking about what kinds of intuition pumps will persuade others to agree with you. Indeed, the reason why people started talking in terms of absolute morality may have been because it's more persuasive to say "God commands you to do X" or "Absolute morality commands you to do X" than it is to say "I would really like it if you did X and really dislike it if you didn't. Pretty please? :D"
Each of the approaches to grounding ethics that Paul cites may appeal to different people. Some feel moved by moral rights, some by utilitarianism, some by care ethics, and some by religious or spiritual messages. It's a matter of choosing the marketing pitch that's most compelling to the target audience. Certain people do feel uneasy with non-religious groundings because they sympathize with the intuitions that Paul lays out, e.g., that nature is red in tooth and claw, so why should humans act any nicer if they're just animals? Needless to say, that instance of the naturalistic fallacy is not an argument; it's just a sentiment that some people find compelling. (I myself find it to be nonsense.) For those who can't easily be persuaded in other ways, probably Paul is right that appealing to religion could help.
That said, I'm wary of doing this too much. While caring about animals is really important, so is not having a deluded outlook on the world. Religious views especially can fall prey to perverse assumptions that make animals get hurt, like the belief that nature is "how it was created to be" and therefore we shouldn't "play god" by intervening. Some of the most passionate defenses of the view that nature teems with joy come from religious believers trying to prove that God isn't evil. I doubt this is what Paul had in mind when he said that religion "has been remiss or complicit in practice, with respect to animal abuse," but it applies here as well as to direct human-caused suffering.