Applying Economic Principles to Other Studies
This paper seems to be about practical policy. It seeks to understand what drives the killing of elephants. While what you write makes sense in theory, there might be some hidden factor that we do not perceive that has a much bigger effect (for example the actors of the market may not be rational) so looking at the data is always welcome.

I would also say that your economic reasoning is not the only possible one. I could also provide a line of reasoning reverses causality 


More elephants are killed (this may be random, or it may be a result of unemployment in other sectors).
Because demand is unchanged, prices fall. 
Since authorities always seize half of the ivory contraband, more contraband is seized.


If this is true then the seizures and the price change is a result, implying a different market dynamic. Unfortunately the paper only tested correlation, not causality, but this highlights that there can be more than one explanation to a phenomenon. In fact it is even possible to argue that the price of ivory and the number of elephants killed is negatively correlated:


More elephants are killed (this may be random, or it may be a result of unemployment in other sectors). 
The authorities, who until now looked the other way, notice this and start seizing ivory, leaving almost none on the market. 
The price of ivory rises.


Which of these explanations you find logical is somewhat subjective. Because there can more than one, a logical explanation may not be the right one. That is why econometric papers (could) provide useful insights.