How did the New Deal affect America’s attitude toward the role of government?
The term "New Deal" was intentionally ambiguous and euphemistic. It suggested that what was being proposed was something like Huey Long's radical suggestion of a "redistribution of the wealth," but it was actually much more conservative than that. The term "New Deal" comes from the game of poker. It suggests that rather than inaugurating a socialistic system, there would be a reshuffling of the cards and then the people could continue operating independently in a free market system. However, the conservatives, and Republicans generally, understood a new deal to be a redistribution of all the country's wealth--taking everything away from the haves and sharing it equally with the have-nots. The New Deal was very popular with the have-nots but very unpopular with the haves. Franklin D. Roosevelt, himself a wealthy aristocrat, was wildly popular with the masses. When he appeared in newsreels in movie theaters audiences would applaud and cheer. But they probably expected more from him and his administration than they were constitutionally able to provide and probably more than Roosevelt really intended to provide.
The New Deal completely changed Americans' attitudes towards the role of government. Before the New Deal, many or most Americans believed that the government had no real role in maintaining the health of the economy or in providing for people who were too old to work or who were otherwise unemployed. The government was supposed to set tax rates and tariffs and things like that, but it was not supposed to otherwise intervene in the economy.
After the New Deal, this all changed. Americans came to accept the idea that the government would be responsible for the performance of the economy as well as for the well-being of the people. We have come to accept this to the extent that even budget cutters in Congress say they will not touch Social Security or Medicare. These are programs that were unthinkable before the New Deal but which are seen as indispensable by many Americans today.