No doubt they dont care about the people, but they do care about their own image.
Their image would be fucked to the point where republicans would be a dead party if this happened. They know this so a deal will happen.
They are crazy, but they are not stupid to this fact. Expect a last minute deal.
They had a confederate flag waving in front of the White House in "protest" of the shut down just the other day...
They don't give two shits about their image.