The politics of this are so easy.
The electorate, which is an increasingly small slice of the overall population consisting only of those who vote, of course, is personally interested in the government because they work for it or collect some benefit from it. A government “shutdown” is therefore an unambiguously and seriously Bad Thing from their perspective. The only question is who is going to be blamed for it.
As CNN notes, when a shutdown occurred in 1995 during the Clinton presidency, the electorate blamed Congress and exalted Clinton, so much so that his “approval ratings” rose and never again fell below 50%.
All Obama has to do is follow that playbook, which he has obviously and consciously done. The feckless Republicans, quaking in their boots at the mere thought of the 1995 precedent, caved in.
It is as simple – and as vulgar – as that.