Does cutting unemployment really equal savings?

As I read this part in USA Today - (link):

The number of people collecting benefits will fall from 10.5 million to 7 million at the end of July if Congress doesn't extend the payments.

About 400,000 Americans are exhausting their benefits every week, saving the government $2 billion since June and an estimated $34 billion through November.

The government is not really saving that much money. Those who couldn't find jobs, will likely still not be able to find jobs and if they do opt for part time employment will still qualify for food stamps and other assistance programs. Those cost money...

Demand on food banks is increasing, some receive tax support, demand on social service agencies in general is increasing, some receive tax support.

All the Republicans are doing is creating a scenario where perceived savings from one area create increased expenses in another...