Thank you for this answer. I am not sure I agree with this, for the reasons outlined below (in case useful information for you, I upvoted and disagree-voted this):

Paragraph 1: Somewhat minor point, I think you may be drawing a distinction without a difference, i.e. extinction being bad because of the effects of it (no future human happiness, flourishing etc) is putting a disvalue on extinction, because it inherently causes those effects.

In an animal context I would put this as: if an animal species has net positive lives, then extinction is inherently bad, and where there is uncertainty, an animal species should be assumed to have net positive lives. I do not have strong thoughts of how certain animal species trade off against each other (I would have to do further research), but my prior is that this is the direction of the sign even if by a small amount, in part because assuming the opposite by default could be used to justify negative things. I agree factory farmed animals likely have net negative lives, but I cannot see e.g. the species of chickens, i.e. any chicken ever, having a net negative life. Beyond that there is also the biodiversity loss and downstream effects of that (e.g. biodiversity loss being linked with increased risk of natural pandemics), and the fact that human's track record with interfering with nature is not great, but I will leave the argument as equivalent to how you phrased it initially.

Paragraph 2: I summarised how I think of extinction of animals in the last paragraph. The quote "if we think extinction in itself is bad, then we should prefer a planet filled with the only first amoebas to one like ours" given the arguments surrounding it in your answer was one that surprised me. Regardless of whether that is true ('amoeba' did not come up when I did ctrl-f of the document linked, so I would have to read it in more detail to try to find the context), it does not seem very actionable. Given the current world we live in (i.e. a world filled with many species that aren't amoebas), saying that people who care about minimizing extinction want to return to that state (given a lot of extinction required to do so) would need further justification. So it seems like less of a relevant comment.

Paragraph 3: As I think the happiness/future value argument can still apply to animals, I still think animal extinction should be seen as bad (although it is less bad than human extinction). I've already mentioned that I think the baseline assumption is that species of animals have positive lives unless their is evidence that they don't. Without data, I would not update on or make decisions based on 'one animal going extinct may increase the number of animals'. I can see that being possible, particularly in the short term, but I could also see that having negative biodiversity and downstream effects as well.

Overall: My main stance on this animal x-risk should be researched or included in the conversation more than it currently is, even if the outcome of that research ends up being 'this should still not be prioritized'. I can see this potentially changing some calculations e.g. how much climate change should be prioritised as a cause area, and which particular interventions for climate change (there are more people working on this than other xrisks, but I would expect that like global poverty there will be neglected areas). After all, climate change is a big extinction risk for animals.

A note: I think 'we' in the first (EDIT: second) paragraph implies more consensus than the actually is, given that I have not really heard this discussed much at all.