There are two important factors in this analysis. One is that his colleagues used an input-output table to estimate the effects of the drop in construction on other sectors. One concern I would have is that a decline in the demand for an input to construction should reduce the price of that input and thereby increase its demand in other sectors. This will not be picked up by input-output analysis.
The second important factor is choice of dates. They look at the period from 2006 through 2010. Over that whole period, construction is a lot of the story. But from 2008 through 2009, it is not as much of the story.
Having said that, I want to believe that construction was a huge factor in the recession. For one thing, it is consistent with Leamer's findings in Macroeconomic Patterns and Stories about earlier recessions. Second, it makes it easier to tell a recalculation story.