The Republican Party does not care about the deficit. It does not care about growth. The singular animating idea of the Republican Party is that it’s very important to have low taxes, especially on rich people.
This is why there were never any votes on the Republican side for any deal to avert the sequester, as long as that deal included revenues.
But President Obama won the election. Any spending cuts the Republicans want to do going forward will have to be balanced with an equal number of tax increases. If the Republicans don’t like it, then they should win some more elections and come back with the votes to do it their way.