That is what the NYT would have told readers if they followed the same practice they do now in talking about opponents of extending unemployment benefits. The NYT told readers that: "the jobless aid measure is one of the last remnants of the Democrats' jobs agenda, which has largely fallen prey to GOP concerns about the deficit."
How does the NYT know that the Republicans are really concerned about the deficit, because they say are concerned about the deficit? Almost without exception, the opponents of the major pieces of civil rights legislation in Congress claimed that they really were not opposed to civil rights, they just wanted the issue left to the states. Would the NYT tell its readers that the opposition to the legislation stemmed exclusively from concern about states' rights.
In this case, many members of Congress who had no difficulty adding to deficits with spending on the wars in Iraq and Afghanistan or with tax cuts largely targeted to the wealthy, are suddenly concerned about the deficit when the issue is unemployment benefits or aid to state governments. While it is possible that their views about deficits really have changed, it is also possible that concerns about deficits are not the real reason for the Republicans' opposition.
Politicians sometimes are not completely honest in the explanations they give for their actions. Reporters should know this. Therefore news outlets should tell readers what politicians say. It should not try to tell readers what their motive is, because the news outlet does not know.