Maybe I should ask why the LPI report uses such a weird number instead?

Take your population of 10,000 rhinos. It drops by 90% to 1,000. The next year it increases by 90% to 1,900. If you did a normal average of the changes, you'd get (-90%+90%)/2 = 0% change. That's a useless number because there has obviously been some major changes between the two years.

If you use a geometric average you'd get sqrt(0.1 × 1.9) ~= 0.436 or roughly a 56% average decrease. This is more meaningful than saying the average change was 0%.

However, you can't use that number to work out how many rhinos there are. It's used to measure an overall trend. If the average decrease gets smaller, it would mean more populations are increasing and less populations are decreasing. Overall, that's good. That's what the number is meant to be used for.

If the average decline next year is 68% then that'd mean we're doing something right because last year the average decline was 73%. What are we doing right? Which populations are better? Which are worse? You can't tell from the average. It's just a general marker of the overall trend.

A "population" here is one species in one place: one herd, one reef, one flock. The LPI takes roughly 36,000 of those, measures how the size of each changed since 1970, and averages the relative changes into a single figure: -73%. So it's the typical trajectory of a monitored population, not a headcount.

What it doesn't tell you: how many individuals, species, or populations were lost, or even what share are shrinking. Around half of the monitored populations are actually stable or increasing. The average is pulled down because declines tend to be larger in magnitude than gains, and because the index is sensitive to how long each population has been monitored (short, noisy series swing harder).

On "why not just sum the animals": you can't add individuals across species (a sardine and a rhino aren't one unit), and any raw total gets dominated by the most numerous taxa, so you'd mostly be tracking a few exploding populations. Per-population relative change is the attempt to avoid that.

Whether it's the best indicator is a fair fight, and the OP's link (Our World in Data, Hannah Ritchie) is the clearest plain-language read on it. The catch is that the headline usually gets reported as a headcount, which is exactly what it isn't.

My own attempt, but I'm confused.

There might be many herds of each species, all scattered in different places. The report counts all the animals in each of these herds over time, and looks at how those change. Consider some rhino's with two herds that are far apart. One of them is a giant herd, with 10000 rhinos, while the other is tiny with just 50. If the giant herd mostly dies out, and drops to 1000, while the tiny herd gets 5 times bigger to 250, then the average percent change is (-90% + 500%)/2 = +205%, even though almost all the rhinos died.

Except this is wrong; they actually take the geometric average, which I think would be sqrt( 0.1 * 5) = 0.25, a 75% decline?

Now, swap which populations do well. the 10000 rhinos become 50000 strong, and the 50 rhinos shrink to just a family or two of 10. The relative percent changes are the same; 500% for one, 10% for the other, so we would still report a 75% decline.