Yes, they could have possibly overlooked things that appear to be obvious in retrospect. Happens all the time because it may or not have been the focus of the paper, or the time constraints may have not permitted this level of research. This is why we usually have multiple papers, each building on each other over time, on the same data sets/topics.

Since housing starts are more likely in areas seeing rapid housing price appreciation, and construction employs many illegal workers, you could make just as strong of a case that increased housing costs drive unauthorized migration rather than the other way around.

That's why it's so hard, and so important, to disentangle correlation and causation.